George Liquor, The Fire Chief and Haggis McHaggis as well.
Disney's classic Mickey and the Beanstalk features Donald Duck near the start, almost delusional from hunger. It's one of his most dynamic moments; some find it hilarious, others quite a bit creepy. See it here.
Eric Cartman. He kills Scott Tenorman's parents and feeds them to him in the form of chili. Why? Because Scott scammed him out of $16. On his page on the official South Park wiki, it has a detailed paragraph on his mental health. Apparently, Eric Cartman shows signs of sadism, Schizophrenia, identity issues, gender confusion, psychosis, psychopathic nature, narcissism, and possibly even insanity.
In addition to Cartman, there is a good cast of antagonists which undoubtedly belong to this trope: Saddam Hussein, The Woodland Critters and Mickey Mouse. But Mel Gibson (in the canon of the series of course) is the best one.
Randy Marsh, especially in "T.M.I." and "A Nightmare on Facetime".
"Hell On Earth 2006" shows the "antics" of Ted Bundy, John Wayne Gacy and Jeffery Dahmer, Real Lifeserial killers. Said "antics," are called so because they become a parody of sorts of The Three Stooges after their initial introduction.
The obviously insane Hat McCullough from "Free Hat", though his defenders claim he killed those twenty-three babies in self-defense.
The Crazy Farmer / Janitor from Beavis And Butthead. Mainly in the episode "Cow Tipping" where he goes chainsaw crazy.
Itchy from The Itchy & Scratchy Show. Itchy is a mixture of Ax Crazy and sociopath, sadistically killing Scratchy in various painful ways, and with different instruments: chainsaws, axes, guns, etc.. And the worst is that it only does it for fun.
Cat from CatDog, in the episode "Monster Truck Folly". Although it could also count as a Crazy Awesome.
In Drawn Together , the mall policeman in the episode "The Lemon-AIDS Walk". At first it seems very relaxed, but suddenly shows its extremely violent when he threatens to kill a civilian woman pointing a gun if Wooldoor refused to throw some candy. When Wooldoor is captured, the policeman killed the women anyway just for the pleasure.
Also, Captain Hero.
Rocko's Modern Life: Rocko and Heffer are on a bus tour of Paris with a bus driver who is determined to keep his passengers on the bus at all costs, so much so that he chases them around the city after they escape.
Wendell from Frisky Dingo is very definitely this. He randomly kills people for no conceivable reason, offers to abort a baby with a machete, then when said baby is born and revealed as a gigantic ant larvae monster, he rides it to the location of a doomsday device to destroy the earth... just BECAUSE.
With his pants off no less, commenting throughout on how wrong it is but how he's going to continue to do it.
Gaz also counts given she once happily used a device that could destroy the world and was DISSAPOINTED when it failed to work. And then there's what she did to Iggins...poor Iggins. He really should have listened to Gaz's warnings.
Darkwing Duck's foe Negaduck is about as close to a kid-safe Ax Crazy as you can get, especially considering it's a Disney show. Negaduck's favored weapon, for example, was a chainsaw, and he takes great delight in attempting to blast cute fuzzy bunnies with a shotgun. This may or may not be a case of being a product of his environment.
Negaverse Launchpad? Happy with firing a bazooka at cute little girls. The Muddlefoot family? Tank's OK as a Non Evil Twin, but Herb, Binky, and Honker were shown with a row of severed duck heads on a pole in their yard. With their beaks sewn shut. Fun for the whole family, that Disney!
Disney one-upped itself in Gargoyles, with Demona. During the City of Stone arc, she manages to turn all the humans in New York Cityto stone, and then is seen roaming the streets, casually destroying humans-turned-to-stone with a huge mace. This one didn't so much bypass the radar as tore the radar a new a-hole. And since it's Disney, nobody called them on it.
Around the end of Avatar: The Last Airbender Azula definitely qualifies, having absolutely lost it after Mai and Ty Lee betrayed her. She has an absolute breakdown before her final battle with Zuko and Katara
Eva and Izzy from Total Drama Island. Izzy does appear as a lovable goofball but she shows signs of this.
Danny Phantom has Danny's evilfutureself as this. The first thing he did upon "reawakening"? Steal the ghostly half of Big Bad Vlad Masters (Our Ghosts Are Different is in full force here) and then kill his human half in front of Vlad, who viewed Danny as a surrogate son. In his first scenes, he kills at least a dozen people as well and spends much of the episode killing or planning to kill some more. The fact that even Vlad refuses to tell Danny how his human half died speaks volumes.
Rampage of Beast Wars flips between this and the Blood Knight. Originally, he attempted to kill anything in sight, no matter whether they were Maximals or Predacons. In fact the only reason he joined the Predacons after his first appearance was because Megatron held his life in his hands by removing his spark and keeping it for torture/incentive for Rampage to serve him, otherwise he'd probably try to kill them all. The fact he's one of the most heavily armed Transformers in the series doesn't help. He even mentions being a cannibal!
Futurama features Roberto, the dangerously insane, psychopathic robot whose literal function is to stab others (there's a reason he resembles a tombstone). His many insane acts includes robbing the same bank three times in a row, killing other robots to provide a disguise for himself, and "practicin' my stabbin"!
Roberto: I'm thinking of a number between 1 and 10. Guess it, and you die first. Bender: Uh....56? Roberto: 56?? 56?! Now that's all I can think about! I'LL KILL YOU!
Then there's Robot Santa.
Zoidberg's race go Ax Crazy during their mating season.
Wanda. Specially when she is mad, demonstrates the sanity that she has, or rather, lacking. Not to mention it also demonstrates extremely sadistic tendencies. Specially in the episode "A Bad Case Of Diary-Uh!".
Heloise on Jimmy Two-Shoes is a malicious Enfant Terrible who loves spreading misery as much as her boss. The only person she doesn't want to hurt is Jimmy, and he's not completely safe from her wrath. Also, Saffi gets this way around statues.
From The Amazing World of Gumball comes Banana Joe, a character who redifines the meaning of the term Ax Crazy. In "The Kiss", he plays it straight, chopping a tree down FILLED WITH HIS CLASSMATES with an axe. All to prevent a kiss which he fails to do. Sort of.
Curly from Hey Arnold! is hinted to being this at times.
The early version of Daffy Duck was usually a fairly harmless nutcase, but in "The Daffy Doc" he loses his job as a doctor's assistant and proceeds to knock Porky Pig out, smuggle him into the hospital, and attempt "surgery" on him with a saw to prove how capable he is.
Roger Smith, in his worst moments, shows traits of this.
Also, the Principal Lewis.
Adventure Time has several of these, with Magic Man, Me-Mow and Ricardio being the most prominent examples of very deranged psychos.
Ice King and Tart Toter are really a subvertion, are obviously unstable characters, but rarely or never are extremely violent, especially Tart Toter, who is not particularly a villain example.
Flame Princess. Whether you think she is good or evil, it's clear she enjoys burning things a little too much, made abundantly clear after going on a rampage in "Vault of Bones".
Lemongrab is a very interesting case. In his first appearances, he was just a guy with true autism, social awkwardness, and a dominant impulse. However, in his last two appearances, he has become definitely in a full-blown Ax-Crazy: almost mercilessly kills his clone, reason? just for break a toy, and in "Too Old", he ends up eating him and abusing his own people for the smallest things. What is scarier than a Ax-Crazy like him? an Ax Crazy in power, becoming in a truly insane tyrannical dictator.
Princess Bubblegum. She can be pretty scary and warlike.
Mac's personality goes from a very Nice Guy to completely lunatic if he gets even the smallest amount of sugar.
Mojo Jojo from The Powerpuff Girls. Often shows various features of mental instability, especially when he's pissed.
Archer gives us Cheryl (who sometimes answers to "Carol", "Carina", "Cristal", and possibly a host of other names). She's a compulsive pyromaniac, gets off on suffering physical and emotional violence, once spent 6 months believing that she was a werewolf, and is more often than not stoned out of her skull on either rubber cement or LSD-laced Gummi Bears (or both.) Luckily for everyone else at the company, she seems to get off more on her own pain than anyone else's. There's also Barry, who appears to have two personalities and has murdered several people, including the entire staff of a space station.