Fist of the North Star has loads of these, Souther probably being the most accurate, based on his childhood-crappitude-to-adult-bastardry ratio.
Sadly enough, this excuse gets completely omitted in the movies, in which he turns into a one-dimensional villain solely in it For the Evulz.
Mazinger Z: Dr. Hell was a poor child that was physically and psychologically abused by his mother, who constantly declared he was an unwanted son and her life would be much better if he did not exist. When he attended school he was constantly belittled by professors accussed him of cheating and beaten by his schoolmates. When he was in college, he took pride on his grades... and then a foreign student (Juzo Kabuto, Kouji's Grandfather) surpassed him easily before wooing the woman he was in love with. After some incidents where he tried making some good action and he got beaten by it (such like saving the life of a little girl and being beaten by her father because the man believed he was molesting her), he fully snapped out, and he declared he was sick of it all, people were mediocre idiots did not deserve living and one day he would purge hit world off idiots, and everybody would kneel before him. And then Nazism happened and he... got tips about how achieving that.
Fushigi Yuugi tosses this in at the moment the villain is defeated, and the heroes suddenly pity the man who's been tormenting and manipulating them for 52 episodes. The villain himself chastens the hero for being so rude as to lay bare his psyche just as he was about to die a mystery.
Seto Kaiba's issues with his adopted father on Yu-Gi-Oh! and Manjyome's issues with his older brothers in Yu-Gi-Oh! GX. The GXprotagonist, Judai, is revealed to have grown up with workaholic parents who didn't have enough time for him, prompting his obsession with the game of Duel Monsters.
Marik is suggested to have turned evil because he was forced to live in a tomb and guard the Millennium Items against all invaders, and because he was forced to undergo a painful tombkeeper initiation ritual. Granted, Dark Marik was something of an alternate persona, but he was one created from the normal Marik's resentment, and normal Marik still wants revenge against Yugi. Yami Marik is the embodiment of Marik's anger and rage. Keeping it suppressed for years, along with the crap treatment he got from his father, caused him to snap and essentially manifest a split personality. This being Yu-Gi-Oh, the second personality is rather real...
They also point out the flaw of logic that Dark Marik resents the Pharaoh considering at the time all he did was be dead and remain dead when it was his abusive father who made his life miserable and forced him to be a tombkeeper. Really the Pharaoh being around at all means his job is officially complete and he's completely free.
Yami Bakura/Thief King Bakura takes the prize among Yu-Gi-Oh! characters, though. He wants revenge for the pharaoh (Not Yami Yugi, his dad) having his whole village slaughtered in order to make the millennium items. He even watched them, while they were still (mostly) alive, be dropped into a pot of boiling gold. Gold boils at a temperature of 2807.0°C ( 5084.6 F), so no wonder they only showed silhouettes. There is a chance too, that he would have had more difficulties due to his purple eyes and white hair. Pale hair, skin and/or eyes in Egypt were often thought to be signs of demons, which is why Kisara is treated the way she is.
Fullmetal Alchemist: Father began his existence trapped inside a flask and used as a resource of alchemic knowledge. His desire to be free and to never again be bound by anyone or anything is what drove him to create the Philosopher's Stone and try to obtain the power of God and create the homunculi, becoming emotionless in the process just because he wished to have a family.
In the first anime, Envy claims one of these due to having been abandoned by his father Hohenheim. The excuse is really not intended as a redeeming factor but more of a retrospective explanation for why he's so obsessed with Hohenheim and the Elric brothers, and its revelation is immediately followed by Envy stabbing and killing Ed.
Scar's turn to evil started after the Amestrian military launched a war of extermination against his people, the Ishvalans. The soldiers who caused the most casualties in the war were the State Alchemists. Scar himself was mortally wounded by the State Alchemist Kimblee, and his brother was only able to save Scar's life by sacrificing his own. Following this, Scar became a Serial Killer of State Alchemists, particularly targetting those who participated in the Ishvalan War of Extermination, in an attempt to avenge his brother and his people.
In Nobuta Wo Produce, the much-vilified bully Bando is revealed to have an abusive boyfriend. Luckily, this is quickly followed by some actual, brilliant character development for her.
This is quite a popular backstory for Naruto villains, anti-heroes, and general douchebags. Neji was bitter and full of rage because he and his father were kept as a servant class within their clan, and his father was eventually killed so his body could be substituted for the head of the clan's.
Kimimaro spent most of his childhood locked in a cage except when he was brought out to fight (only in the anime).
Sasuke takes a turn down a dark path of vengeance because his older brother killed the rest of their family and Mind Raped him, twice.
And Gaara... oh, Gaara. It's hard to tell which was the worst part of his childhood: being hated by everybody in his village, being constantly targeted by his father for assassination, not being able to sleep for years on end, being betrayed by the one person he thought cared about him, who reveals he actually always hated Gaara right after trying to kill him, or being infused with a demon spirit while still in the womb. Not so surprising he ends up being a serial killer. What's even more surprising is that, once he gets over it, he's actually a pretty nice guy.
And Naruto himself has an excuse- he grew up alone and friendless, shunned by everyone in the village for (to him) no apparent reason- he just doesn't use it. Except for the occasional Bart Simpsonesque minor act of vandalism very early on, that is.
Spoofed in Naruto The Abridged Series, when Sakura tells Sasuke she's not interested in Naruto because he's got such a downer background, yet isn't using it as an excuse to be emo.
And Itachi Uchiha, the Well-Intentioned Extremist. His childhood was during the great ninja war and he was traumatized by it (Quote Madara: "For a child, war is hell."), so he grew up to be a man who values peace above anything else. Above anything else. And then, you know the story. Then again, unlike some of the other Freudian Excuse-inspired villains, he was forced into the situation, albeit because Danzo knew about that part of his personality, and did not turn evil because of it.
Danzo's excuse was revealed to be not being picked as the Third Hokage when he was a rival with Sarutobi in their youth. The exact situation is that the Second Hokage named Sarutobi when the latter volunteered to be a decoy for 20 enemy ninja when a mission went bad. Danzo volunteered so as not to seem weak, stating his grandfather and father died in battle (saying that it's the way of shinobi life), but the Second recognized that Sarutobi was the best pick for successor and left them to be the decoy.
Orochimaru has an indirect excuse. His parents died when he was young and over the years his desire to see them again led him to study forbidden jutsus, whether to directly revive them or gain immortality to wait until they were reincarnated. Sasuke mocks him for having forgotten the very purpose for all of his crimes.
There's also the former Dragon Pain real name Nagato. He wants to make the ultimate weapon of mass destruction and bring suffering to the entire world because he saw his parents killed by enemy ninja and grew up orphaned in a country constantly embroiled in both civil wars and war with other countries. When he, Konan, and Yahiko lead a peaceful organization, they're attacked by Hanzo, backed by Danzo, both of which have brought a small army with them thinking the group are revolutionaries. When Nagato is left with the choice of killing Yahiko, the leader, or watching Konan be killed, Yahiko grabs Nagato's knife and kills himself with it.
Interestingly enough, unlike many other Freudian Excuse villains, his most despicable acts in the eyes of the fanbase (killing Jiraiya, destroying Konoha, and almost killing Hinata) occurred after his Freudian Excuse was revealed, almost as though to counteract any sympathy we might feel for him when he expands upon it afterwards. As some fans believe, while Pain has a tragic backstory, it isn't enough to justify the extent of his plans and actions.
Kakuzu; started out life as Shinobi loyal enough to his village that he willingly let them experiment on him, then when chastised for failing to assassinate the First Hokage (perhaps the most powerful ninja to date), he turns on his superiors and chooses to only put faith in money.
Sasori everyone! He grew up without parents; grandmother Chiyo never broke it to him that they were really dead, instead claiming they were away on mission after mission after mission. So he fills the gap by creating two puppets to replace them, but quickly realizes that wooden dolls can't take the place of real parental love. His emotional growth is stunted to that of a young teenager, which makes sense because he preserved his body to look that of his 15-year-old-self. And so begins the journey of a cold, heartless, stoic Puppetmaster who cares little of kidnapping ninjas and turning their bodies into his personal puppets and weapons. There seems to be a trend of broken children in the Sand Village...
Kabuto's excuse is that Danzo brainwashed his mother (the person who'd helped him develop his identity after amnesia and who he emotionally relied on) into not recognizing and trying to assassinate him, and the only person who then comforted him after that and offered to help him find himself after that was Orochimaru.
And it was pretty much inevitable Tobi/ Obito would get one too. He watched Rin be killed by Kakashi after being almost fatally injured and was inspired by the real Madara to create a world of illusion where there were "no heroes".
Played with in JoJo's Bizarre Adventure: Dio Brando was raised in poverty by an alcoholic father who treated him like dirt and worked his mother to death. He left for the Joestar manor to take their money as his father requested on his deathbed. Dio, however, did so for himself, not because his father (whom he despised) asked him to. Even Speedwagon didn't buy into his Freudian Excuse; he was raised in similar circumstances to Dio, and he could tell Dio was a bad apple just by his smell.
In Gundam 00, it's probably easier to point out which characters do NOT embody this trope rather than the ones who do.
Arlong first appeared as a racist who terrorized innocent people for money. But as it turns out, him and fishmen in general have a pretty good reason for hating humans for what they did to them (which includes a long history of slavery, oppression and discrimination).
It's not just Arlong being a fishman, it also turns out he was raised in the Fishman District, which was an orphanage meant to house the enormous amount of children orphaned by the slave trade and the chaos caused by the Great Pirate Age on Fishman Island. Said Fishman District quickly deteriorated into a lawless ghetto, not the best place to grow up in. On top of that, the only person Arlong ever looked up to - Fisher Tiger, was a former slave who was killed by the marines for the crime of freeing slaves and who was betrayed by the humans he helped. He also spent some time in Impel Down, which "cruel and unusual punishment" doesn't begin to describe. It kind of messes with the viewer; Arlong is one of the few villains in One Piece that successfully and directly kills a character onscreen, yet he also is the only villain that is actually shown crying of sadness. To top it off, he's crying over the death of a mentor, something usually reserved for the Straw Hats themselves. Adding more fuel to this trope, Arlong Park was modeled after Sabaody Park - a place fishmen and merfolk were unable to attend because of the high risk of being kidnapped - and that Arlong, Hachi, Kuroobi and Chew longed to go to as children. Hachi even says that what they truly wanted was to be part of the human world, even if they took it too far.
Played with with Hody Jones, who's even more of an anti-human extremist than Arlong, with none of his redeeming qualities such as treating his fellow fishmen with respect (he's perfectly willing to use his own crew as human shields, plans to commit genocide on any fishman even considering friendly relations with humans including assassinating the queen of Fishman Island, who was the biggest advocate of peace, and ultimately tries to destroy all of Fishman Island, including his entire crew.) When he's finally asked what the hell humans did to him to make him this way, his response is nothing. Humans never did anything personally to him. He's just the product of growing up in an environment with fierce anti-human sentiments.
Nami subverts this trope. One would expect her to hate Fishmen after 8 years of involuntary servitude to Arlong and the Fishman Pirates. Keep in mind that Arlong is the guy who killed her mother Bell-mère in cold blood right in front of her when she was 10 years old. It was never touched upon whether or not she harbored any hatred for Fishmen during those 8 years, but hundreds of chapters later in the manga (specifically Chapter 627), Nami states that she never hated Fishmen, at all. She only hated Arlong, because as an individual, he was so insufferable and vile. She was also very distrustful of Hachi when they met up again, but after seeing how remorseful he was about the things he had done as a former member of Arlong's crew, she forgave him. Nami was also visibly disturbed by all of the racist comments the slave auctioneers were making about Hachi in Chapter 502.
Moria was once like Luffy, idealistic and adventurous, but his entire crew was wiped out by Kaido, leaving him as the sole survivor.
Crocodile has this heavily implied when concerning his desire to kill Whitebeard, likely because he may have wiped out his crew, and caused him to become cruel and sadistic.
In Black Lagoon, the Creepy Twins/Psychos For Hire Hansel and Gretel have a definite Freudian excuse for being so deeply, deeply broken, creepy, disturbing, sociopathic and just plain wrong in the head. Born in Romania and abandoned by parents too poor to raise them, they were raised in an orphanage no better than those of 19th century London (a sadly common story during Nicolae Ceausescu's rule). Then they were sold to a producer of kiddy snuff porn, where rape, torture and murder were part of their daily routine - to the point where they can no longer grasp the concept of surviving without killing. Unlike many cases of the Freudian excuse, this one actually works, making the twins both the most evil characters in the show, and the most tragic.
Villain Protagonist Natsuo Ishido from Teppu starts off as a fairly standard example of the Jerk AssRival trope, but as the story goes on, it's eventually revealed that her cold, sadistic demeanor is at least partially due to being bullied as a child by her classmates, as well as being physically abused by her older brother.
Aion from Chrono Crusade starts his plan to overthrow the demon's government and destroy the world when he discovers that Pandaemonium used to be a human woman and she was pregnant with twins when she was transformed—twins that just happened to be him and his brother, Chrono. That's just in the manga, however—in the anime, he's just evil for the sake of being evil.
In Princess Tutu's second season, it's revealed that Rue's reason for becoming Princess Kraehe is that she believes she's the daughter of the Raven and he's told her that only the Prince from the story could ever love her.
Actually, to be more precise she was kidnapped as a child, brainwashed and raised as Princess Kraehe and became Rue to forget and be happy for a while..
Blood+ does this masterfully with its antagonist Diva. Her 'childhood' was so full of torment, neglect and abuse that you "almost" want to give her hugs and love and tell her it'll all be alright - almost. Then she does something completely Ax-Crazy and unforgivable like raping and murdering the protagonist's 12-year-old little brother and is just as terrifying and squicky as before, only now it... kind of makes a little sense. She also steals his face.
It's stated in Hana Yori Dango that the reason Domyoji Tsukasa is prone to semi-sociopathic fits of violence (including beating the crap out of and having the rest of his high school torture a student who accidentally squirted lemon juice in his eye, and beating another student to the point that his organs ruptured) is that his parents ignored and neglected him in order to focus on running their vast corporate empire (ironically, the reason he was able to get away with his gratuitous exploits). He's not all bad, however...
Rolo Lamperouge from Code Geass... whoo boy, where to begin? He starts off as a Tykebomb sent to spy on and eventually kill Lelouch, until Lulu uses the feelings of brotherly love Rolo's built up over the past year to manipulate him with the full intent of killing him once his usefulness runs out. This proves to be effective because Rolo, as a raised-from-birth assassin, has never known any form of real kindness. Unfortunately, this backfires on Lelouch when Rolo becomes so obsessive and fixated on being the only person his "brother" cares about that he murders Shirley, one of the few people Lelouch was genuinely close to, and plans on killing Nunnally, who's Lelouch's whole reason for living. Amazingly, Rolo managed to be Rescued from the Scrappy Heap by saving Lelouch from the Black Knights' mutiny at the cost of his own life, declaring that though he knows Lelouch has been manipulating him the whole time, his own feelings of love were genuine, and for the first time in his life he's doing something he wants to do instead of something he's been ordered to do. Cue Tear Jerker music and Alas, Poor Scrappy effects.
He's not the only one, either. The absolutely vile Psycho for Hire Luciano Bradley, according to the light novels, was abused by his evil father and neglected by his alcoholic mother, and killed his father (who said out loud he didn't care for his child's life) at age seven. Also, Emperor Charles and his brother V.V were witnesses of all the backstabbing, hatred and death surrounding the Britannian Imperial Family ever since they were young children, which culminated when they saw their mother get a carriage dropped on her.
Lelouch's mother is mysteriously killed and his sister Nunally is crippled, and he and Nunally become political hostages in Japan. They are assumed dead once Britannia invades anyway. Which instills in him a hatred against Britannia and a murderous intent against his own half-siblings and father.
More of a subversion, considering Lelouch has sound political reasons for wanting to overthrow the empire, even if his desire to do so STARTED from his mother's assassination.
A lot of his motivation comes from the opposite of what his father does - his father thinks that those who are weak are useless, and thus Lelouch (likely due to disgust with his father) incorporates the antithesis of his father's views into the Black Knights' slogan.
Suzaku has one, too, though not quite as bad. As a child, his father, Genbu, would have let the war go on until Japan was destroyed, so Suzaku killed him, resulting in almost immediate peace. His guilt over the incident and complete lack of punishment for it is why he argues that anyone who uses the wrong means to achieve freedom (in short, anyone who doesn't bow down to their conquerors and earn it through their system) is wrong. It gets a little hypocritical when he starts doing the things he speaks out against, while still calling others out for the same. Midway through the first season though, it is revealed that this is more of an excuse for him to put himself at risk in his pursuit of a so-called honorable death on the battlefield, in order to absolve himself of his guilt over going unpunished for killing his father.
Mao in Code Geass. The guy was a six-year-old orphan when C.C. gave him his Geass, which ruined any chance of a remotely normal life. And then she deserted him when she was the only person he loved. * sniffsniff* BOOHOOHOOHOO!!!
C. C. was an orphan at the time she received her geass. Her wish? To be loved. Said wish/geass becomes meaningless when Power Incontinence sets in, and with everyone fawning over her, she is not truly being loved for who she is. It only gets worse from there, as the nun who gave her the geass reveals she was just using her, as she passes on her code and dies, leaving C. C. with immortality and, feared by many to be a witch, having died quite possibly several thousands of times, an emotionless, cynical, distant shell of a human. It's little wonder why she has used other people in escaping her immortality, and in her skewed view of compassion, tried convincing them not to become too friendly, dependent, or trusting with her in order to protect them. (The Mao example, in particular; she loved him too much to allow her fate onto him, and wanted to shoo him off in order to make him more independent. Too bad it backfired spectacularly.) Ultimately, she overcomes this, if not the immortality, thanks to Lelouch's intervention.
Most bad Code Geass characters have Freudian Excuses of some sort. The attention they're given in the actual show largely depends on their relevance to the characters' situations.
Many GOOD Code Geass characters seem to possess this trope at first as well, but escape it by either having massive amounts of character development, which contradicts the trope, or actually exploring the depth and legitimacy of the excuse.
In the manga version of Rosario + Vampire, Hokuto has one of these. My father beats me and monsters at school beat me as well, so I must destroy the world and recreate it in my image!
Subverted in YuYu Hakusho. Sakyo says that his childhood was exactly like that of his four brothers, who went on to become civil servants, but he became twisted for an inexplicable reason and sought out brutal pleasures until he eventually joined the Black Book Club; as he says, "The depravity up here began through no one's fault but my own". Mukuro, however, plays it straight in that she was adopted and sexually abused by her stepfather, and sought to take out her anger on the world by killing people, which is what led her to become one of the three strongest figures in the Demon World. However, in the manga, Hiei notes that she had a hypnosis-instilled manufactured memory of her adoptive father showing her kindness, which was made to come to her mind every time she wanted to kill him, and she has "grown strong because [she's] confused, not because [she] suffered". Keiko once wonders if Yusuke turned out the way he did because of how irresponsible Atsuko is.
Younger Toguro's students were murdered by a powerful demon Kairen and he was badly beaten; in order to kill this demon he willingly gives up his humanity to become a power hungry mass murdering demon.
Czeslaw Meyer from Baccano!, whose reason for wanting to kill all the passengers in the dining car of a train in order to root out any other immortals who might be a threat to him stems from being tortured for two hundred years by his guardian, who he had trusted.
He struck back a lot sooner than that, but it was still a long period of the kind of torture where they do things no one could survive, Healing Factor activate, repeat.
Inverted with Ladd Russo: A tragic happening in his past (only very slightly) less of an Ax-Crazy lunatic - he's pickier about his targets now.
The Funimation dub of Dragon Ball Z did this with Vegeta during his death scene on Namek, having him justify being evil by blaming Frieza for taking him as a child, saying "He made me what I am." As well as revealing that a good portion of this involved forced servitude (in other words, Frieza forced Vegeta into serving him because he threatened his father, and when he did everything for Frieza just to keep his dad alive, Frieza killed him anyway).
Broly technically has one as well, as apparently one of the reasons why he went Ax-Crazy was because of traumatic events relating to his birth - mostly Goku crying, although there was also the attempt on his life by King Vegeta (having one of his guards execute Broly by stabbing him in the stomach just because he feared his power level), and then being threatened with death by Freeza when he destroyed Planet Vegeta, and it is later implied in his first movie that he also destroyed several planets due to Paragus' influence (not to say he didn't destroy any planets before getting that control ring on him, but he certainly started destroying more planets than before as a result completely against his will).
Vision of Escaflowne's Dilandau Albatou is crazy due to horrific government experiments being performed on him, changing him from a sweet little girl to a psychotic pyro killer. The ruining of his face and destruction of his Dragon Slayer force by Van only exacerbate his issues.
Luxus, one of the biggest Jerkasses in Fairy Tail prior to his defeat and Heel-Face Turn anyway used to be a cheerful kid who loved his grandpa Markarov. That changed when Markarov kicked Luxus's father Ivan out of Fairy Tail for endangering his comrades. After that Luxus resented both his grandfather for valuing his True Companions more than his family and the concept of friendship that close and strong in general. Since, as mentioned earlier, he's one of the biggest assholes in the series, who even went so far as to threaten to destroy the guild's hometown if Makarov didn't hand it over to him because he wanted to make Fairy Tail stronger, it's not a very good excuse.
From Death Note, we have Misa Amane, a Serial Killer who generally shows no signs of sympathy for her victims nor hesitation to kill them. These people can be anyone who gets in her way. Her excuse is that her parents were murdered before her eyes in a robbery. This makes her support the original Kira, Light Yagami, who punished her parents' killer after he escaped conventional justice. This explains, but does not excuse, her behavior.
We've also got Teru Mikami, an even crazier prosecutor turned serial killer whose victims include even the unemployed, eventually. However, as a child his mother died, and he endured horrific and near-constant bullying at school, simply for defending his classmates' honor. He was beaten up and disillusioned about justice in middle school. Then, four bullies ran over his slightly unsupportive mother and all five died, prompting him to believe that justice was real. He wasn't sad that so many died, just relieved, and he decided that when he was older, he would work to extend the arm of justice to protect people everywhere. Somehow, that led to him becoming a prosecuting attorney, and then a vigilante. Pretty screwed up, huh?
He also believed he had psychic powers to wish the unworthy dead (he thought so similarly to Light, that the people Light killed often ended up being people Mikami thought should die), even before he got a Death Note and actually had them. He also repeatedly refereed to Light/Kira as "God" and as such saw himself as "doing God's work".
Also in the second film, Light dies in Soichiro's (who, by the way, does not die here) arms, begging him to believe he acted as Kira to put TRUE justice, which the latter had told him about since childhood, into effect. Not much sympathy toward Light comes out of this excuse....but a LOT toward Soichiro does, as he breaks down crying once Light passes away after saying this.
In Glass Fleet, Vetti's horrific actions (including raping the main character, Michel) are all forgiven in the end and he gets a pass. The reason? He was sexually and emotionally abused as a child.
Each volume of Saki begins and ends with a page from an omake story about Yuki, who travels to the Land of Tacos to find that some unnamed king from a neighboring territory has... um... harnessed the power of corn to wage war—leaving behind none to make new tacos with—because his parents divorced when he was young over how bad tacos tasted.
Mahou Sensei Negima! toys with the whole idea in regards to Evangeline. Evangeline was turned into a vampire at age ten, inadvertently killed her parents during the transformation, and was essentially considered an undead abomination for the next 600 years or so, causing hundreds of people to try to hunt her down and kill her. Despite this, she claims that she's utterly evil, refusing to acknowledge the fact that her (very long) Dark and Troubled Past may have influenced her actions a little. Asuna points this out, noting that Eva isn't really as evil as she thinks she is, and that she's just acting that way in response to all the crap she's been through.
Scarlett from Steam Boy is an utterly spoiled snotty brat, treating Ray with haughty disdain and punching her little dog in the head when it tried to run away from her. Then later on she tells Ray that she has five 'mothers' - one that cooks her food, another that takes her shopping, another that helps her get dressed, another that reads her stories at night and another that teaches her, and angrily tells him that she doesn't run off to them if she's in trouble (unlike Ray, who was worried about his). It's obvious at this point that Scarlett is so bratty because she's never had anything close to a parent, just servants who did everything they were told. She does get better by the end of the movie, though.
Wiseman from Sailor Moon uses Chibiusa's feelings of inadequacy and abandonment to turn her evil.
He also twists Prince Diamond to his purposes by preying on the prince's feelings of rejection and bitterness at the Moon Kingdom.
In Medaka Box, Medaka, who only thinks the best of everyone (up to a point), believes this must be the reasoning with all the delinquents in her school. For instance, she thinks the thugs using the kendo hall must be a heavily disillusioned team when in fact they really are just thugs who happened to find a large empty building. Nonetheless, her sheer force of will combined with the thrashing she and her partner give them causes them to become a real kendo team.
In Romeo X Juliet Lord Montague gets one of these dumped on him in Episode 17. "The What Do You Mean, He's Not Likable?" kind....
Why is Tetsuo from AKIRA not on the list yet?! After enduring years of kids bullying him and trying to make him cry, and being the youngest member of a biker gang who is looked down on by everyone, including his best friend... he discovers he has power greater than all of them. And... cue his Roaring Rampage of Revenge.
All of the members of the Team Rocket trio have Freudian Excuses. Jessie was raised poor and her mother was killed in an avalanche, accounting for her Tsundere personality. James was raised by neglectful parents who wanted him to marry a Yandere named Jessiebelle, accounting for his timid and repressive demeanor. Meowth never had any family and learned to talk just to impress a female Meowth who still rejected him, accounting for his conniving features.
Pokémon Special. Lance's excuse for wanting to Kill All Humans? As a little boy, he sees a Dratini dying in a pool of industrial sludge. As he hugs it to his body, he can feel its pain, see its thoughts. He sees through its eyes that Humans Are the Real Monsters, and that must've been pretty damn traumatic for him as well as the Dratini.
Doesn't his Blackthorn heritage come into it somewhere, too? Clair's just as off-the-handle as Lance, just less...ambitious.
During Bleach's Soul Society arc, most of the Soul Reapers who defended Rukia's attempted execution that weren't part of Aizen's plot had one of these. Soifon was bent on ensuring the laws were upheld due to her pain of being left behind by Yoruichi, Yamamoto was firmly dedicated to enforcing the laws of Soul Society despite his personal feelings, Komamura felt he had a life debt to Yamamoto and followed him, and Byakuya had sworn an oath on his parents' grave never to break a law of Soul Society. Byakuya is an interesting case, as he had also sworn to his dying wife, Rukia's sister, that he would always protect Rukia. This made Byakuya, who seemed to be the coldest of the antagonist Captains, to actually be the most conflicted.
Rurouni Kenshin IS this trope. Essentially every character with a combat role has some sort of melodramatic horrific past. To name a few:
Kenshin was orphaned, briefly enslaved, raised by a violent (if brilliant and essentially decent) jerk, made into a murderer by a government he later came to resent, and wound up accidentally killing his own wife. Though that last is more his excuse for being angsty and standoffish, since it's ultimately the reason he got through the war sane and swore the no-kill vow.
So(u)jiro(u) was an illegitimate son who suffered constant abuse, killed his entire stepfamily in self-defense around eight or so, and under the wing of a megalomaniac grew into a prodigy assassin with no feeling other than 'contentment.'
Yukishiro Enishi, the brother of the wife Kenshin accidentally killed, was so traumatized by witnessing the event that his hair and eyes turned white and turquoise respectively. And, since he'd already hated Kenshin, consequently spends the next ten years plotting revenge, becoming a blackmarket weapons kingpin, and nursing his sister complex.
Friendly young monk Anji turned into revolutionary death monk after the local head man had his temple burned down with all his orphans inside, to curry favor with the young Meiji government and their imperial Shinto project. (This was a real thing. Orphans optional.)
Aoshi's original one was he didn't get any chance to fight and win honor in the war, and now the government doesn't want to hire ninja, and while his second one "I got my men who loved me killed by my jerk employer for no good reason".
Kamatari wants Shishio to love him, but he's not as womanly (or awesome) as Yumi and doesn't have the combat skill to earn his respect, so he just does what he's told. Why he's a scythe-wielding crossdresser who thinks a taken, hideously scarred maniac like Shishio makes a good One True Love is never touched on.
Sano actually started the trend, with Kenshin outright asking him what had messed him up so bad when he was obviously a decent sort. Comes out that as a kid he became the only survivor of the Sekiho Army, a company of irregulars betrayed by the revolutionary government, and is really mad and disillusioned about it. Defeat Means Friendship for Kenshin, though, and Sano becomes his new best friend.
Saito is gloriously unexcused, though, as is Shishio and Jin-e.
Chou too, actually, who appears to be just a cheerful Kansai psycho who really likes swords.
Surprisingly the Gag Dub of Shin-chan gives one for Penny's and her mother's violent behavior by saying that her father abuses them. In the original no explanation was ever given for there violent outburst.
Not a villain, just a Jerkass, but Kanda Yu of D.Gray-Man turns out to have a very good reason for being so aloof and cynical. Namely, this, over and over again for months. Additionally, he was created artificially and knows it, grew up with very little human affection, and was eventually forced to fight the closest thing to a friend he had to the death. However, other characters with incredibly traumatic backstories normally turn into The Pollyanna instead.
It gets worse. The Second Exorcist project, which Kanda is a product of, involved implanting the brains of dead Exorcists into new bodies, and he eventually started regaining memories of his own death. Kanda's life sucks so much that using an Exposition Beam to show someone his memories qualifies as Mind Rape.
In Elfen Lied we have Lucy and Mariko. Lucy is harassed by her peers at what appears to be an orphanage because of her horns. They even killher dog. Mariko, on the other hand, is nearly killed by her father at birth. His wife convinces him not to before she dies. Even then, Mariko is forced to live in a solitary confinement chamber where she can't even move. And she ends up killing the one person that ever talked to her.
All of the Diclonii hate humans and turn psychopathic, but they arguably have a good reason for this. They're almost invariably hated by their peers for their horns, if they're even allowed to live outside at all. Most of them are, from birth, used as experiment subjects. However, many of them have a very strong instinct to kill humans. Nonetheless, Nana proves that not all of them are as Always Chaotic Evil as people make them out to be.
All human villains in Rave Master get one of these. King wants revenge on Haru's dad for getting his wife and son killed. Lucia, King's actually not dead son, wants revenge on the world for killing his mom and for being locked a prison from the age of 6 to 16. Doryu wants revenge on 'the light' for being condemned to darkness by the humans of a town he tried to create to dissolve racism (he failed). Reina wanted revenge for her father's unjust torture and death. Jegan lost it when the woman of his dreams chose someone else. Shuda and Julius didn't have too good of reasons, but Shuda was more of a good guy on the wrong side anyway, and Julius was... special.
Subverted in Durarara!!, where Shizuo is more than a little distraught that his angerissuesaren't the result of abuse or childhood trauma.
Shizuo: Why did I turn out like this? It wasn't my family. There was no childhood trauma I can think of. I never watched any violent anime or read violent manga. Didn't watch movies either. So that leaves only myself, doesn't it? It's gotta be me, right?
Ginias Sahalin from The 08th MS Team is ultimately revealed to have tremendous underlying abandonment issues thanks to his and his sister's mom walking out on them as children, which he copes with by using said sister and later his super weapon with extremely Freudian implications as replacements for her.
Referred to in Soul Eater, when Stein tells Medusa of the doctors who wanted to know what trauma in his childhood could have caused his desire to experiment/cut people up. He claims there wasn't one. Medusa herself has so far not given any excuse for her terrible actions, other than the fact she feels like it, and a world under the authority of Lord Death just isn't appealing to her. On this much, she and Stein appeared to be in agreement to some extent.
Crona is given one. After all, his/her mother is Medusa. Who wouldn't be messed up after being raised by somebody who experiments on you with the black blood, forces you to kill cute little bunnies, gives you an illustrated book on various methods of killing, and locks you in a closet without food for several days if you don't do what she says?
Actually, this trope initiated the series. It was originally just a one-shot, the chapter we now know as "Maka", about a girl who worried her best friend would leave her for a more attractive partner. This was due to the fact that her parents were in the process of getting a divorce because of her father's repeated cheating. The audience liked it so much they demanded more. The rest is history.
Ax-Crazy Alois Trancy in the second season of Kuroshitsuji. Somehow his family was killed, after which he and his little brother had to live alone. Then his brother contracted a demon, who killed him along with everyone else in the village. This left him alone and easy prey for the old Earl Trancy, to whom he was made a sex slave, and all this before he was fourteen.
Ax-Crazy genocidal maniac Gennai Doma from Black Lion swore to wipe out all ninjas off the face of the earth because a group killed his wife and nearly killed him.
Oskar von Reuentahl from Legend of Galactic Heroes is not a villain but one of the "good guys" (although he does a sort of Face-Heel Turn eventually, but even then he remains unfailingly loyal to Reinhard whom he admires and respects). However, behind his calm and composed facade he's a bitter misogynist, an increasingly broken and self-destructive man. He was born with heterochromia, and his mother took his brown eye as an omen about her illicit relationship with her brown-eyed lover so she tried to gouge it out when he was still a baby. After this she went insane and eventually committed suicide. This drove Reuentahl's father to alcoholism and emotional abuse, often telling his son that it would have been better if he had never been born.
Odin of Kenichi: The Mightiest Disciple has a pretty stupid one (but that's the point): He fought Kenichi for a yin-yang pin when they were kids, and lost, but Kenichi felt sorry for him and tried to give it to him anyway. At first it looked like it was the influence of the Great Sage Fist that made him crazy, but we see in a flashback that, if anything, the Fist stabilized him for a time.
In Jackals, minor antagonist Daryl the Mincer is afforded one of these; the local crime syndicate's information broker attributes Daryl's homicidal tendencies to being sexually abused by his stepfather as a boy.
A large number of the mysteries in Monster involved the elaborate traumatic backstory of Johan prior to his becoming a serial killer.
Claymore: Opheliais the wayshe is because she saw her older brother get devoured by an Awakened Being and was driven to insanity in order to avenge him. The Awakened Being turned out to be none other than Priscilla who in turn has her own Freudian excuse for her behavior because she saw her family get eaten by a yoma who - just to add to the trauma - had killed her father and had taken possession of his body.
In Flameof Recca, Kurei hates Recca for just being born. Kurei was the Hokage heir, until his younger brother Recca was born. Two children of fire in the same generation is thought to be a bad omen, and Kurei was sentenced to death, and very nearly murdered for supposedly having an evil flame. Very incorrect, as it turns out Kurei has the most sacred flame that can be produced. He was fully innocent. Recca was the cursed flame holder. He was however spared at the last moment. But, he and his mother were forced to live on the village outskirts where they were ostracized. Oh yeah, and Kurei lost his entire clan and mother in a war in a single night.
And if that wasn't bad enough, when he arrived in the future, and was taken in by a kind woman, he was abused, mentally and physically by her corrupt husband and conditioned into an emotionless weapon.
In Fruits Basket, every character that's constantly in the strip has a Freudian Excuse. Tohru, the main character, had her mother killed in a traffic accident and her father when she was only three, which caused her mother to ignore her completely for a period of time.Kyo, her boyfriend and later husband, had his mother commit suicide, and his father hated him as a result. He believed that it was his fault for most of his life.Yuki has a rather extreme version: Akito, who probably has the greatest Freudian Excuse of all of the characters, locked him up in a room for most of his life, and beat him. He's shown with a whip in the anime. It goes on and on, even extending to minor characters like Kakeru and Machi (who becomes not so minor when she and Yuki start dating). Subverted by Kimi, who had been bullied by some girls when she was younger. They told her that she should just use her looks to get boys to do what she wants. She follows their advice.
In Revolutionary Girl Utena, even Akio has one. Back when he was still Dios, he worked himself almost to death while trying to save people, so his sister hid him inside their house. When people came looking for him, she tried to stop them, and she was stabbed many times, turning her into the Rose Bride. It is implied that this event also led to Akio and Dios becoming separate people.
A subtle example in Bio-Meat: Nectar. After the first timeskip. the newest member of the True Companions notices that the rest of his new friends are all vegetarians, a fact which surprises him. The reason? In the first story arc of the manga, the meat that is used as a substitute for animals escapes and kills all their peers and families.
In Saint Seiya some of the God Warriors from the Asgard saga had one in their respective backstories: For example, Alioth Epsilon Fenrir hates and mistrust humans after his parents were killed by a bear, he was abandoned, being rescued by a pack of wolves. Also, there was Benethosh Eta Mime, who had an abusive adoptive parent. Mime kills him after he told him that he was the murderer of his biological parents. However, it all turned out to be a Thanatos Gambit from his adoptive father, who wanted to both atone for the accidental killing of Mime's family and die in battle instead of his long-time illness, so he put on a Jerkass Façade for many years..
Kyo Koi O Hajimemasu: Kyota's distrust and manipulation towards women who accept his advances stem from childhood trauma. When he was still a child, his mother left their family with a man she had an affair with. His father, being quite violent, began physically abusing him after that. His only way out was a nurse he liked talking to as a substitute parental figure...until the day he caught her cheating on her husband in the infirmary. This made him believe women would only seek to use and betray him.
In Sekirei, Mikogami Hayato is revealed to be a Lonely Rich Kid who spent his life wishing for something "amazing" to happen that would "change and destroy" him. His wealthy parents ignored him unless he acted out, and his intelligence made it seem impossible to connect with his peers, leaving him feeling hopeless. He describes his childhood as being born into a "narrow world" controlled by others, and apparently spent a significant length of time living alone in the United States after angering his parents. Sending a child to live alone on the opposite side of the planet seems to have been their version of grounding, with them having the Old Retainer deliver a message warning that they intended to do it again when he got into a fight at school. Then he got his wish and found something amazing, when he met Mutsu and became part of the Sekirei Plan. He declares that he intends to use the power granted to the victor to change the boring, restricting world into something more exciting.
During Noir Altena, the series Big Bad, is portrayed as an Evil Mentor who doesn't care about anyone or anything except her goal of bringing the blood soaked days of Sodat's past back again. This goes in direct conflict with the modern Sodat society whom are more like politicans. During the final episode when the usually calm Altena is going through her Villainous Breakdown, a flashback is shown of Altena as a little girl in a war torn town. Her parents are dead and an enemy soldier finds her. It's implied that the soldier kidnapped her and used her as a sex slave.
Weiss of RWBY harbours a strong distrust of Faunas. This is because the White Fang, a terrorist group that fights for Faunas rights, has terrorised her and her family for years. The White Fang have kidnapped several of her friends and family, and it's implied that she was a victim of her father's temper brought about by the White Fang's actions. The thing is, a later scene reveals that Blake is a Faunas and a former White Fang member, as she blurts out after a heated argument. Weiss' Fantastic Racism is wrong, sure, but it's easy to sympathise with her if her backstory truly was that crapsack. The whole aforementioned scene is quite a tragic example of two victims of a very morally grey conflict butting heads.