The original Earl of Lemongrab is the living embodiment of this trope. Impulsive, psychotic, irrational and violent when he is in a bad mood, and has even committed cannibalism.
The Ice King is mentally unstable and likes kidnapping people. However, he appears to draw the line at using lethal force on sentient beings, even though it's repeatedly shown that he's more than capable of it.
Though Ocho's explosive temper was previously established, in "The Uncle" he comes off as a psychotic delinquent who starts fights unprovoked, pretends to kidnap someone as a "prank", and was implied to have wanted Gumball to beat Darwin with a hammer just to prove his friendship with Ocho was more important.
Barry Robinson, when he's not medicated. However, his true personality isn't completely held off by the meds. Even with his usual "simpleton" demeanor, he occasionally slips out a psychopathic thought or two.
Roger, full stop.
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.
Around the end of Avatar: The Last AirbenderAzula 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
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!
Katz. Just check out some of his killing methods, and smiles.
Freaky Fred as well. The fact the only thing he'll murder is your hair does not help as much as it should.
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.
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.
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'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.
In Drawn Together , the mall policeman in the episode "The Lemon-AIDS Walk". At first he seems very relaxed, but suddenly points a gun at a woman and threatens to kill her if Wooldoor refused to throw some candy. When Wooldoor is captured, the policeman killed the women anyway just for the pleasure.
A number of the Drawn Together housemates qualify for this, with Captain Hero, who destroyed his home planet out of spite, being the most obvious.
Vicky. She's a crazed psychopath who will go above and beyond to torture kids For the Evulz.
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!" when she finds out that Vicky called her a "fat pink squirrel".
Mac's personality goes from a very Nice Guy to completely lunatic if he gets even the smallest amount of sugar.
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.
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"! Not that it's truly his fault; whatever idiot built him programmed him to be a sociopath.
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.
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.
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.
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.
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.
Mr. Cat from Kaeloo. He owns a large number of weapons such as bazookas, chainsaws, guns, and the like, and spends half of each episode usingthem on Quack Quack.
Ren himself. He's usually prone to psychotic fits of fury for little to no reason at times. He's even willing to kill anyone. This is not Played for Laughs in some episodes, however.
George Liquor, The Fire Chief and Haggis McHaggis as well.
Rick from Rick and Morty. He has been shown on multiple occasions to go absolutely berserk and overboard upon provocation, including ripping off the equivalent of a robot's testicles and shoving it down said robot's throat when tipped off to the police, engaging in cross-dimensional crossfire with himself when he thinks that other versions of him are trying to kill him, literally going ax-crazy and violently murdering various clones of himself (including a child and an infant), exploding an entire alien colonisation for putting him in a simulation and creating accurate genitalia for his grandson Morty, destroying an entire universe in order to spite its creator, programming his space-car to torture and murder in cruel and horrific ways or manipulate/coerce people into staying away from it when advised and committing countless other atrocities.
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.
Aku from Samurai Jack was always a sadistic, extremely homicidal creature, but in "Jack and the Zombies", he's very much the living embodiment of this trope. Notice his maniacal cackling while fighting Jack. During the beginning of the fight, Aku's laugh is absolutely psychotic, and then he goes on a murderous rampage.
Aku: "Oh Samurai? Where are you Sa-mu-rai? You can run but you cannot hide, CAUSE I CAN SMELL YOUR BLOOD!"
Groundskeeper Willy is just one hair trigger away from becoming this trope.
Sideshow Bob started to qualify for this when his attempts to kill Bart included trying to have him cremated alive and when he visibly removed his cell mate's face and cut off his own. He also ties up Bart while singing casually about wanting to gouge out Bart's eyes. And then there's the Treehouse of Horror XXVI segment "Wanted: Dead, Then Alive" which has several moments that are genuinely disturbing. First, Sideshow Bob finally kills Bart. And it's not an over the top, comically ridiculous death, but fairly realistic. Bart gets harpooned through the heart, and since that doesn't immediately do the trick, Bob RETRACTS the harpoon and RIPS HIS HEART OUT. He then tears out Bart's large intestine and dances around with it and singing about how happy he is that Bart's dead before using it to tie his corpse to himself like a harness. It's also then revealed that Bob has kept Bart's corpse as it starts to decompose, eventually stuffing him in a compartment in the wall, even talking to him and playing golf with his body like he was alive. To make it worse, Bob revives Bart and then kills him again in various other ways—including mauling him by a lion and bludgeoning him with a sledgehammer while he's still conscious.
There was also a one-shot (well, technically, two-shot) character employed at the power plant in "Marge Gets A Job." This guy was shown polishing a shotgun and talking to himself:
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 Earth Day Brainwashing Society, 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.
Kaynar: At least I'll have the voices in my head to talk to.
Eva and Izzy from Total Drama Island. Izzy does appear as a lovable goofball but she shows signs of this.
Mal, evil personality of Mike, of the second generation. He just loves inflicting pain on others.
Scarlet true self is this. To the extent where she tried to blow up the island which would have killed everyone.
Josse, of the Ice Dancers from Total Drama Presents: The Ridonculous Race, is this. Aside from throwing temper tantrum just about every time she and her partner Jaques aren't in 1st place, she has no qualms about hurting people, even if the person in question is her own partner.