A lot of Sluggy runs on this, though it doesn't always have to be associated with a character so much as the whole story's premise. And there's also Bun-bun. Now, he doesn't actually act crazy or necessarily do very crazy things, but his entire premise is that he is theoretically the strip's cute talking animal, being a small, cute rabbit and all, but contrasting with this and turning it on its head ridiculouslyBad Ass and very evil. At first that's just a joke, but it soon elevates him to Crazy Awesome as he is shown to be one of the coolest characters around.
As for stories that are Crazy Awesome, how about "Holiday Wars"? What starts out with jokes about an evil rabbit trying to kill Santa Claus develops into a years-long arc that goes through Santa starting to go insane and creating an Easter Bunny Terminator to go after Bun-bun, then Santa being infected by alien DNA and trying to take over the world only to be stopped by Bun-bun and elves with nerf-weapons, and finally escalates with Bun-bun threatening to take over the world by killing many of the anthropomorphic personifications of the various holidays—such as the Pumpkin-Headed King of Halloween and the Alien Santa—to steal their powers. And it's as cool as it is insane.
The seemingly normal toaster that shoots toast over a dozen feet into the air through walls. He once tried to sell it to a hardware company, but when the CEO tried to try the "Leggo My Eggo" trick fingers were lost. Then he suggested selling it to the military.
Largo, Dom, and Ed from Megatokyo. Actually the whole series could be crazy awesome, but those three look crazy even to the other inhabitants.
Jayden and Crusader's resident mad scientist Smic is known for his crazy awesome moments including, but not limited to, harnessing the power of sunspots to fill the house with pizza, firing a cannon using cream as gun powder and defeating an alkaline super-monster using his bare hands. He is also extremely British.
Dude Guy in the Nsider comic Shy Guy Tale was regarded by fans as the most awesome thing ever. This was lampshaded in one comic by Dude Guy poofing away a would-be attacker, and replying "I'm just that awesome" when asked how he did it.
Heat Man: Alright. As a token of my awesomeness, I'm going to give you twenty seconds to answer my questions. 1) Who in the name of ME are you? 2) In what way are you NOT a complete Me-damn fairy? And most importantly... 3) Why are you not on your knees and bowing before my greatness? You have twenty seconds, lower being! Get a-answerin'.
Freeze Man: Oi oi, just twenty? That seems like a very low amount of time to be "a-answerin'" your inquiries. 1) The name I am known by is BATMAN. 2) I don't say 'HEY! LISTEN!' every 3.1415 seconds! 3) Probably because I have a limited amount of poses, and so I don't have a 'bow down before your greatness' pose.
Katarakis from Starslip. He tried to take over the world with art, visited the formation of the universe, and destroyed the timeline and the site itself with a spork.
Both Killroy and Brandon in Killroy And Tina, which is probably why they get along so well.
Also Belkar, from the very beginning. Sexy shoeless god of war indeed...
Xykon has some aspects of this as well. Dude tossed a bouncy ball with a Symbol of Insanity spell inscribed on it into a room full of paladins. The outcome contained a balance of Hilarious and Horrific worthy of the Joker.
Everyone in The Adventures of Dr. McNinja lives and breathes this trope. Start with Dr. McNinja, who runs a medical practice with a gorilla for a secretary, regularly fights hundred-foot-tall giants, punched out Death, air-surfed a robotic Dracula, battled laser-eyed cyborgs, saved the world with tennis, and massacred countless clones, Sky Pirates, and Mooks.
Then you have the supporting characters — his entire ninja family, his raptor-riding 12-year-old sharpshooter sidekick, his mentor the Ben Franklin clone, Dracula himself, the time-traveling ex-astronaut Mayor, his college superhero team and his motorcycle. The only person who isn't crazy awesome is his ex-girlfriend, and she once skyjacked an airplane while dressed up as an ancient Incan robot.
Having noticed his own reputation for creating such characters, the author finally said "screw it" and created King Radical. He's a king of the "cereal mascot" variety from an alternate universe called the Radical Land who became the leader of the Cumberlandnote Population: 21,518 mafia after sealing away the evil unicorn king Sparklelord. He routinely rides his motorcycle over alligator pits, has trumpeters on skateboards announcing his presence, and eats only fresh, locally produced organics, prepared simply and with passion.
How about Joe Chaos, from Another Gaming Comic? He's notorious for holding onto anything—anything, including cursed items—and using it against the DM later, such as using the slightly-broken rules for choking to give himself time against a poison attack with a cursed Necklace of Choking. On top of all the rest, he's also Crazy-Prepared, ridiculously intelligent, and recieved Training from Hell in the art of playing Dungeons & Dragons. His nickname comes from the time he used a cursed Gauntlet of Rust to completely negate a kyton—a demon that attacks with metal chains. Joe's name isn't even Joe. It's Irving.
Also from AGC: Nuclear Dan, who can solve any problem by fireball and concussion.
Homestuck: Bro. Like Dave, he has a katana too, but prefers to kick Dave's ass with a different weapon: Li'l Cal. And when said ass-kicking is over, he flies away on a rocket-skateboard. In a series that runs off the sheer absurdity of the characters and situations, it really says something when this guy is the Crazy-Awesomeest of them all.
Bro Strider's Alpha universe version, Dirk Strider, never fails to deliver Crazy Awesome. In the course of a single flash video, [S] Synchronize and [S] Unite, he manages to revive both of the Alpha universe heroines from death with a kiss, create two players' kernels (while importantly failing to prototype them), travel through dream worlds and time, and orchestrate a plan that hinged upon his death by decapitation and his crush kissing his severed head, transported from 413 years in the future, to bring Dirk from the dead. All while riding a rocket skateboard. And, presumably, rapping.
HatGuy from xkcd. He may not always count, But he has his moments, Like when he became Secretary of the Internet just so that he could turn the Capital Rotunda into a giant ball pool. Or when he commandeered a Russian submarine so he could go get his hat back.
Randall Munroe himself. The guy made a map of the internet. Twice.
Mob Ties: Sidney Burns and his booziness were what got this whole mess started, have continued to propel him through much of the story, and is actually a vital part of his character. The fact that the booze is self medication to keep the trauma of several months being a prisoner of war from turning him into a rage filled monster only makes it more awesome. Of course, there are a few side effects.
That the guy frequents a bar called the Trainyard that is full of crossdressers (and not the pretty kind) because he likes the margaritas, throws bottles of beer with devastating accuracy to silence loud students in class, renamed an entire yakuza clan after a high school mascot, and improved his boxing skills by training with a giant panda. Oh, and he often communicates via puppet.
Last Res0rt has Adharia, who is easily just a (bi-curious at best) Fish out of Water otherwise... except for the fact she carries a freakin' Efreet around her neck and can order it around.
Max Facepuncher, as with everything else, takes it to eleven, introducing himself to the main character by jumping out of a plane that's doing a swoop at high speeds, then immediately starting a fight. That's pretty much how he says hello.
Axe Cop: The comic rivals Dr McNinja in Crazy Awesome territory due to the fact that the story is based on the imagination of the artist's six-year old brother (which also makes him Crazy Awesome, as the comic is quite popular).
Yet Another Fantasy Gamer Comic has Captain Fang, the Dread Pirate. He either utterly insane, or have near-terminal case of dyslexia which he doesn't try to circumvent and is easily distracted. He's also either absurdly lucky or a good enough pilot to go into tricky waters, get even with a crack crew in an artillery duel (while sitting on the cannonface to the breech) and... somehow... got into semi-final of a great tournament. All the while rambling and apparently not paying attention to anything.
The entire main cast of Narbonic. Helen Beta Narbon, mad scientist who went insane in an Italian restaurant: nearly a dozen people died from the monsters she created using just what she found in the ladies' room. Mel Kelly, intern and gun nut, who in another timeline took over the world in order to destroy it. Dave Davenport, minion, who is too good at fixing things, including the Kill Sat death ray he built from what turned out to be a broken mail sorter. Artie, the superintelligent gerbil whose good-natured attempts at making things better have led to an uprising of robot copies of a rival mad scientist, and a pack of evil hamsters.
He went to Canada to investigate missing U.S. soldiers and boinked the town sheriff, who handcuffed him to her bed. Just before she and everyone in the town turned into werewolves. At which point he escaped and saved his temmates. His teammates are an unstoppable fast zombie and a genetically engineered battledog, and he got the Big Damn Heroes moment. And he got an I Call It Vera moment at the same time.
Madeline the paladin from Rusty and Co.. She's a "Ditzy do-gooder". How ditzy? The first time we meet her, she wields a hoe and is convinced it's Holy Avenger — because the gnome who sold it to her said so. Only to return later with a spade she called vorpal halberd. She also frequentlyfights with closed eyes. Which somehow makes her attacks more precise, allows to deflect thrown daggers and apparently lets "magical" agricultural equipment work as advertised — paladitz did beat a tavern-ful of various armed creatures and a pirate crew with that hoe, while the spade was used to the same effect as if it was a vorpal halberd. And she can SMELL evil.
Exterminatus Now gives us Morth The Betrayer, who, while normally not in this category, manages to gain this status due to an unusual tactic: not only will he summon a demon lord and ascend to a demon himself, he'll gain thousands of new followers for his master as well. How? By broadcasting the entire thing online.
Sydney of Grrl Power has the ability to substitute confuse for any attribute when making a skill check.
Kiel'ndia of Drow Tales has definitely crossed over to this. She has always been a little out there (she treated the readers as her imaginary friend). After the timeskip, she's weaponized us. The readers are now manifest as a swirling cloud of black demons apparently only she can see, and she uses us to fight other characters and capture demons.