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.)
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 liked to recreate the slaughter of his own family whenever he got bored.
Darla wreaked some very bloody havoc of her own.
Faith. She later does a Heel-Face Turn, and both reforms and becomes more stable.
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.
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).
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 kill off his own race when they tried to stop him from make the Daleks, told the Doctor that if he were giving 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 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 is an example of someone who has gone through Ax Crazy moments while still retaining her protagonist (and woobie) status. Her main violence has been against bad guys, but 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."
Heroes: Gabriel Gray, better known as Sylar. Take one case of validation issues and compound with superpowers and murder, and you have a doozy.
I Carly: Nora. She also held a medieval hatchet that scared off the trio back to their incarceration.
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...
Kamen Rider Ryuki: Takeshi Asakura aka Kamen Rider Ouja who was a sadistic serial 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 last season there's an inmate named KillCrazy. Ironically, at the first chance to actually kill something, he runs headlong into a doorframe and knocks himself out cold.
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, 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.
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.
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.
Aaron's son 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".
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.