Babylon 5: Centauri Emperor Cartagia, epically ax crazy and in charge of a planet. Prone to laughing and doing a merry jig with you one second and casually having you executed the next. Planned to have his world blown up by the bad guys in exchange for his ascent to godhood (or so he believes.)
The Sniper, in "Ceremonies of Light And Dark". He took seven days to kill a Minbari. He cut his victim to pieces, starting from the digits and extremities and tying off the body part to be removed next, so as to avoid his victim dying from blood loss.
Hector "Tio" Salamanca. In flashbacks, he is revealed to have been a particularly sadistic member of the Juarez Cartel who possessed a pronounced lack of empathy towards those outside the organization and was horribly abusive towards his own family (even while professing to value them above all things). In this manner, he served as the role model for his Ax-Crazy nephews, Tuco and the Salamanca Twins.
The Cousins. They kill almost everyone they come across. They also have a literal ax.
Drusilla. Drusilla's craziness is a result of the mental trauma she experienced as Angelus tortured her and made her watch while he tortured and killed her family.
Angelus, who sometimes to recreate the slaughter of his own family whenever he got bored. His progeny Penn later took on the same custom.
Darla wreaked some very bloody havoc of her own.
Faith. She later does a Heel–Face Turn, and both reforms and becomes more stable, although is it sometimes hinted that how her initial mental state lies just below the surface e.g. she is tempted to kill Angel in a hell dimension (so he would no longer place her and others in danger) and working as The Mole she loses it and tries to kill Buffy.
Dark!Willow and Vamp!Willow very much embody this trope, however even normal Willow slides into it occasionally. The comics foretell this will lead to Buffy killing her, and lately she has an actual axe (the Scythe) that's used to slice up Angel's son.
Connor ends up Ax Crazy after killing his godlike daughter Jasmine, snapping completely after learning a man he stopped from committing suicide would have left his family alone, holding a mall hostage and rigging it with explosives, insisting that a man stop holding his daughter wrong (because Connor broke his arm). Connor had some problemswithhisfamily.
Caleb and Warren Mears.
Glory. Juuuust a little bit.
Simone is this, partially because she is psychotically obsessed with killing Buffy. Remember when Faith was like that? Simone's worse.
In the comics one also sees the two half-demons Nash and Pearl. They regularly kill whole groups of humans, or even demons, simply because this makes they both happy.
Michael: "If I let him out of my sight, that's when he starts killing everybody."
Charlie Jade: 01 Boxer, at least in the Betaverse, and arguably the Alphaverse. He seems to manifest a different personality in each world; in the Gammaverse he's a loving husband who is horrified at what he becomes in the other two universes.
Criminal Minds: Most of the killers, as the show tends to only deal with the most extreme criminals. They range from the technically sane but extremely sadistic and heartless to people with almost no grip on reality whatsoever.
"The Ice Truck Killer" and the Trinity Killer. Dexter himself is not Ax Crazy. He's very particular and methodical about who, how, and when he kills, and limits his victims to murderers who have evaded justice (He does do in a couple of child molesters, as well).
Dexter's dark passenger is Ax Crazy. Dexter is obliged to kill from time to time to pacify the dark passenger, otherwise it will take over and make him Ax Crazy. One time Dexter goes longer than usual between kills we see this begin to happen, though he never completely loses control.
Though far less dangerous than the other Big BadsMiguel Prado's descent into vicious insanity is probably the most spectacular so far.
The Daleks. Ex-ter-min-ate! Extra credit to the Special Weapons Dalek. Basically, a Dalek with a BFG strapped to the front. The only problem is that the radiation drives the Dalek insane, and as liable to destroy its own Dalek comrades as the enemy. The Daleks keep it chained up and call it "The Abomination".
Their creator, Davros: who killed off his own race when they tried to stop him from making the Daleks, told the Doctor that if he were given the power to end all life he would do it just to prove that he could, and later on made a bomb that would DESTROY! REALITY! ITSELF!
The Master; he likes to kill whether it helps his plans or not. The best example of this comes from "The Sound of Drums", when he orders the Toclafane to kill one-tenth of the population, just because he likes the word "decimate" (to reduce by one-tenth).
The Silurians of the double-episode "The Hungry Earth/Cold Blood", though it's the military caste due to an unintentional incursion. This apparently gives them the excuse to attempt to start a war despite numerous protests that they don't need to.
Hindle from "Kinda." It's clear pretty early on that he's not very mentally stable, but then he gets worse...and then he rigs up those explosives...
The Cybermen are so sane and rational that they've gone out the far side into a new universe of Axe-Crazy.
River Tam may or may not qualify. Her main violence has been against bad guys; though she's attacked fellow crewmates in the past, but has not done so with enough force to kill. Still, it's enough to freak out more than a few people who are not familiar with her history and motivations for violence.
Particularly memorable is the scene in "Ariel" where she slashes Jayne's chest with a butcher knife because he was wearing a Blue Sun T-shirt, and because he would later try to turn her and her brother Simon in to the Alliance for the reward money. With her psychic abilities, she most likely sensed something stabworthy about him.
Ax Craziness is the hat of the Reavers. As Zoe puts it in the pilot episode, these cannibalistic once-human horrors will "rape you to death, eat your flesh, and sew your skin to their clothing — and if you're very, very lucky, they'll do it in that order."
The Flash (2014): Jesse James, the Trickster. While plenty of the Flash's enemies aren't exactly sane, the Trickster doesn't cause chaos and death as part of any grand plan, or for money, like the others - he's just doing it because he finds it fun. Notably, the comicbook version of the Trickster isn't Ax-Crazy, but the show's version is largely an Expy of The Joker. (He's even played by Mark Hamill, who's voiced the Joker a fairnumberoftimes.)
Ramsay Bolton, even after being somewhat toned down from the books, still manages to be one of the most frightening examples ever.
Gregor Clegane, often called 'The Mountain,' is sadistic, freakishly strong, and brutal in his attempts to sate his bloodlust, with a dangerously short temper and utter lack of regard for long-term consequences. There are a plethora of examples to list that show just how crazy he is. The list includes burning off half his brother's face when they were kids, killing a woman's children then raping and murdering her with their blood still on his hands, trying to kill a jousting opponent he lost to and his own brother when he interfered in the fight, personally picking out which war prisoner would be tortured to death each day in Harrenhal, murdering prisoners as part of his exercise routine, murdering every Northen captive in Harrenhal before leaving as a final "fuck you!" to the Stark army, even though the nobles that could have been ransomed for money, without caring about the fact that he is setting a precedent and the Starks could very well retaliate by executing their far more numerous Lannister prisoners. Also according to his younger brother Sandor he once "killed a man for snoring." This is taken Up to 11 in later seasons, when he becomes a pseudo-undead monster that can rip a man's head off with little effort.
Joffrey is a sadistic sociopath with little regard for human life, but he's also a Dirty Coward so he prefers to use a crossbow on helpless victims. In one of his first scenes without the supervision of another Lannister, the sweet prince begins to cut Mycah with his sword and threatens to gut Arya before Nymeria intervenes. He proceeds to get much, much worse.
Mad King Aerys was so ax crazy that it brought about the downfall of his dynasty, at which point he plotted to incinerate King's Landing out of spite.
Biter proves the truth of his name when he attacks the Hound with just his teeth.
Myranda quickly proves herself to be just as psychotic and insane as Ramsay.
Heroes: Gabriel Gray, better known as Sylar. Take one case of validation issues and compound with superpowers and murder, and you have a doozy.
iCarly: Nora. She also held a medieval hatchet that scared off the trio back to their incarceration.
I Didn't Do It: Sherri in "Dance Fever." She repeatedly threatens to kill Lindy just so that she can be the only person with a perfect attendance record, and she has infected fellow perfect-attendance wielding students with diseases for that same purpose. She's also convinced that by having perfect attendance, she'll become a goddess, specifically the goddess of attendance. She memorized Lindy's schedule and route so that she could place a juice box on the steps she walks on in order to make her fall down the stairs and break a few bones, which would force her to miss school. Later, after Lindy avoided slipping on the juice box, she causes Logan's date to the dance to trip and fall down the stairs so that she could be his date in her place, get into Lindy's house, and make her miss school in any way possible. She even sends Lindy a video message in which she decapitates the head of a doll resembling Lindy!
Lindy: All right, Sherri, what's the deal? I mean, first you come after me at school, now you come after me at my house...when does it end?
Sherri: It ends when you miss school and I become Atenda, (Scare Chord) the Goddess of Attendance! Lindy: Over my dead body! Sherri: Suit yourself...
Insecurity: N'udu is a more restrained version of this; he knows how to kill people using anything and once cleared a long check out line at the grocery store by describing how he would kill the other occupants of the line using the contents of their basket. Like everything in this show, it's Played for Laughs.
JAG: Serial killer Charles "Charlie" Lynch in "Goodbyes". He actually confronts Harmon Rabb with an ax...
Jessie: While in her first appearance, "Creepy" Connie is much more creepy than crazy, she is full blown Ax-CrazyStalker with a Crush in her next appearance. The episode deals with her attempting to get the role of Luke's lover in the school play. How does she do this? By taking out her competition, of course! The first girl she pushes down a flight of stairs, putting her in a full body cast, she welds a wrench to the next girl's braces, and attempts to crush the final girl with a sandbag!
Connie turns it Up to 11 in "Creepy Connie 3: The Creepening" including hiring an actress in order to dupe Luke. In order to avoid paying said actress, Connie kidnaps her.
Connie herself confirmed she's had at least one police line-up.
And her uncle confirms her multiple(!) restraining orders in "Creepy Connie 3"
The kids think Jessie herself is axe crazy in the Halloween episode.
Kamen Rider Ryuki: Takeshi Asakura aka Kamen Rider Ouja who was a sadisticserial killer, he also is the one with the highest body count of the series, he also loves to commit as much destruction as possible and has an absolute blast while doing it.
Law & Order: There have been many episodes dealing with ax crazies.
Keamy is already not a nice guy, but like several people on the freighter it appears that the Island has made him unstable. Extremely unstable. To the point where he casually kills the captain of the ship just to prove a point and intentionally puts the crew (well, the ones he hasn't killed yet) at risk...just so he can complete his mission of getting Ben. He also saves his own life by kicking a grenade at his second-in-command when it lands at his feet.
In season 6, a clearly-not-all-there Claire.
MythBusters: Parodied in the "Mega Movie Myths" episode. In the silent film-esque intro, Jamie is chasing Tory with a pickaxe after Tory forgot to put away his drill. "Heeeeeeeeeeere's Jamie!!"
Oz: Many a inmate is this. Beecher becomes this after taking a level in badass, and some of the guards show these tendencies.
In the eighth season there's an inmate named Kill Crazy. Ironically, at the first chance to actually kill something, he runs headlong into a doorframe and knocks himself out cold. Which saved his life, oddly enough.
The Dwarfers used to occasionally run into Simulants, Ax Crazy robots who were bred for a war that never happen, and bent on destroying all life. Plus the one time they were sent a replacement for Kryten, Hudzen-10, it turned out to be Ax Crazy as well!
In "The Plague Dogs", Ray Kinsey. He used to be sane, and he and his daughter Lilah took refuge at a carnival. Then Lilah stepped on a nail, got tetanus, and he could only watch helplessly as she died, because the medicine they had was taken by raiders. In response, he trained a group of attack dogs, and he didn't react well to Team Matheson killing one of his dogs. In fact, he stabbed Maggie Foster in the leg and hit an artery. Then he kidnapped Charlie Matheson, tied her to a chair, positioned a crossbow in front of her, and rigged a wire to the trigger and the door. She tried to reason with him, and even though she reminded him of his daughter Lilah, he ultimately just left her.
In "Sex and Drugs", Drexel the druglord, who is introduced to the audience by threatening to kill Miles in cold blood just for showing up on his property—only for him to reveal that his gun isn't loaded and laugh his head off. He doesn't get any saner.
"Who the HELL...put bouillon cubes in the shower head!?! Huh? Hm, did you do it? Hm? Did you? If it happens again...I will wait in my SUV, blast me some speed metal — five point one surround sound, heavy on the bass — and someone...will be getting...mowed...down."
Many Meteor Freaks. Bruno Mannheim, a gangster who appeared in Season 8's "Stiletto" was also visibly nuts, twitching in every one of his appearances, and gleefully killing his boss for wanting to go straight. And then there's LX-3, an utterly murderous clone of Lex Luthor, who tries to kill or hurt every single person we see him speak too. He burns the other clones to death while ranting about how "There Can Be Only One", attacks Tess, kills the whole staff at Cadmus Labs, and, to top it off, ties Lois to a scarecrow and sets the field around her alight after placing a bomb in the Daily Planet. All this so he can force a Sadistic Choice on Clark and have a few laughs before The Last Dance claims him.
Richie: Not as crazy as I used to be, still crazy enough to take an eye out.
Other mobsters with a extremely homicidal behaviour definitely qualify... or with a temper that would not have any mentally balanced person. Ralph Cifaretto, Feech La Manna, Johnny Boy Soprano, and Phil Leotardo are straight examples. Even Paulie "Walnuts" has traits of one, but more downplayed compared to the other examples, though.
Captain ("LORD!") Garth from the episode "Whom Gods Destroy", especially apparent when he nonchalantly tosses a planet destroying bomb to one of his henchmen for amusement. Not helped by his delusions of becoming "Master Of The Universe".
Red Jack from "Wolf In The Fold", naturally.
Similarly, the entity from "Day of the Dove" makes its victims Axe-Crazy, so it can feed on their negative emotions for sustenance. The entity itself is a case of Blue and Orange Morality.
Lore, who sacrificed an entire Federation colony to the Crystalline Entity, simply because the colonists were uncomfortable having an unstable sociopath with superhuman strength, speed, and intelligence freely roaming their colony. He would later try to feed the Enterprise to the same entity for no particular reason.
Jamie: You mimsy bastard, Quisling leak fuck! ...Okay, okay, okay. You, Julius Nicholson, being of sound mind but with a body that looks like a giant sex toy, did knowingly do us up the shithole by passing confidential information to the enemy! And I am gonna have your guts as a skipping rope, and your lungs sun-dried and turned into a little fucking waistcoat!
Aaron Echolls may not have used an axe in the show, but his eyes in the rearview mirror scene in the season 1 finale, locking the title character in a fridge and setting it on fire (intending to burn her alive because she refused to give him back tapes proving he had murdered another teenage girl by bashing her skull in with an ash tray (also for refusing to return incriminating tapes), as well as the frequent abuse of his son and the beating-almost-to-death of his daughter's boyfriend, qualify him as a member of the Ax Crazy group.
Logan Echolls has been shown to possess similar tendencies and was described as 'psychotic' in the first episode, during which he bashed in the headlights of Veronica's car because she had inadvertently caused his own to be taken away by getting him arrested. He is shown to have a short fuse throughout the series, although the show ended before he could properly follow in his father's footsteps.
Wiseguy: Vinnie Terranova stumbles across a corrupt toxic waste-disposal company, whose boss ends up beating a woman to death with a golf club. He later turns out to be suffering from mercury poisoning ?- he was "as mad as a hatter".
Xena: Warrior Princess: Callisto. A pre-Heel–Face Turn Xena destroying her village was her Start of Darkness and she blames Xena for everything she's done, but nobody in the series is more gleefully murderous than her, including pre-Heel–Face Turn Xena, and it's clear that she's the way she is because she likes it. Then she gets her hands on Playing with Fire powers a few seasons in, and it gets a whole, whole lot worse.
Boris ends up like this after merging with his vampiric reflection, appalling even the vampires around him with his willingness to ignore their few moral codes, culminating in him murdering the Grand High Vampire so he can steal his crown.
In the season 3 finale Sethius quickly proves utterly insane, killing the vampires around him purely because he can. He intends to replace humanity with vampires because he wants them to run out of food and descend into violent chaos. He's bad enough that Ingrid and the Count actually follow the plan instead of scheming so they can take him down.
The Young Ones: In "Flood", Vyvyan has created a potion that cures people of not being axe-wielding homicidal maniacs. To avoid confusion he has stored it in a coca cola can. Neil has a silly premonition that someone will accidentally drink it later on. Yes, someone does.