In Project Horizons, a Fallout: Equestria fanfic (a fanfic of a fanfic), most of the villains have these. This is lampshades several times, but the funniest is when Blackjack rejoices upon finding a villain who *doesn't* have one, because she wouldn't feel sorry about the ensuing fight.
"The Beyond Birthday case made so much more sense now, BB wasn't a megalomaniac he was L's spurned and neglected lover. Clearly the jam was just the aftershock of being forced into sex with someone that looked vaguely like a crack head. And she had prevented him from burning himself alive, if she had to have sex with L she might have tried to burn herself alive too. Naomi noted to herself that perhaps it was a good time to go to L.A.'s prison and apologize to that poor madman."
In A Charmed Life things are not as perfect in Light Yagami's life as he pretends them to be—he is a very unhappy person and he thinks of Ryuk as his first real friend. Of course he's still a mass murderer...
In The New Retcons Elly's hatred of motorcycles stemmed from being rejected by her first boyfriend and the father of her first child, who was an avid motorcycle rider. Her obsession with cleanliness was caused by her grandmother calling her unclean after she had a child out of wedlock, a feeling made worse when her first marriage dissolved. That same incident caused her brother Phil to swear that he'd never have kids.
In Heir Draco Malfoy is regularly tortured by his father via the Cruciatus curse. Also there's Harry himself- in this fic his abusive childhood is played much more seriously and as a direct result he's now a dark lord in training. Then there's Voldemort— as a child Tom Riddle had a psychotic break when the other kids at the orphanage killed his baby snake in front of him.
The Jaded Eyes Series is another fic where Harry's childhood is played much more seriously as it becomes his Start of Darkness when he snaps and murders the Dursleys. Then he ends up in an Orphanage of Fear where he's regularly bullied by the other kids and has the scars to prove it. It gets worse when Harry discovers that his parents are still alive and have a perfect little family and a little brother who was living a perfect life while he was living in that Hell.
In Diagnosis Harry expressed his annoyance at this particular trope when telling Hermione about his lessons with Dumbledore sixth year.
...He made a point to mention to her that it was hinted that Riddle had been abused, and that his mum had bewitched his father who was a Muggle, and even that he was more than likely unloved and abused, but that it was no excuse for what he became. The abused aren't predestined to become the abusers, and it always hacked Harry off when people, including the Headmaster made it seem like it was some sort of excuse.
Mega Man: Defender of the Human Race subverts this. While Dr. Wily does have an abusive father who colored his outlook on life, the story takes pains to show that Wily himself is a horrible man who deserves punishment.
Stray discusses this - it's set in the Metal Gear universe, land of villains with unhappy childhoods, and at least two characters (including one of the protagonists) are former Tykebombs. However, Adamska (one of the tykebombs in question) eventually rejects the idea that a person's essential nature can be changed by the actions of other people.
In Gensokyo 20 XX, we have a rather non-villainous example in Yukari, in that it be a fairly safe assumption some of her issues she's had, especially wanting to have a child to such an extent that it tears her up inside and some issues with attachment (either would be described as "something to love, something to hold, occasionally scold, and something that would take away an intense pain"), seem to stem from being abandoned by her parents, to whom she referred to as "two large shitbags", when she was two.
In Yu-Gi-Oh! Forever, several villainous characters have a Freudian Excuse for why they turned to the dark side. 1) Jan Shimizu, the Black Sheep of his family, recounts late in the story that his father, Ryou Bakura, frequently treated him with subtle contempt because Jan's appearance reminded Ryou too much of the abusive spirit of the Millennium Ring. It led Jan to leave home and change his last name, to distance himself from the family, and led Ryou to declare I Have No Son concerning him. 2) Michael Bronte, a member of the JUDAS organization's Quirky Miniboss Squad, explains that the reason he joined the group was initially to exact revenge on his parents—specifically, his stepfather used to physically abuse him as a child, while his mother turned a blind eye. Present-day Michael is an Ax-CrazySlasher Smiler who enjoys perpetuating Cold-Blooded Torture on his victims. 3) Angel Snow, a member of the Zodiac terrorist group, grew up with a drunken and physically abusive adopted father. The Dragon then liberated her from that home situation, and she jumped at the chance to give him her loyalty. 4) James Truesdale, another Zodiac member, has a particularly painful one that he hides behind a mask of stoicism and thinly-disguised contempt for his sisterand father: his mother died giving birth to his sister, and his father made him keep his grief hidden for the sake of the little girl. He blows up spectacularlywhen he finally reveals his past and his repressed anger.