I'll list the traits I can think of.
His most general trait is that he's a heckler.
He also is a magician-he can drag his hole around, drop in it and then appear behind the antagonist.
Then - he is a specific kind of heckler-mostly calm and cool when in control of the situation-but when he sometimes is not in control, he becomes really irritable and his pride is sorely wounded which just makes him lose even more control.
Bugs and his Warner Bros. cohorts were a revolution in cartoon personalities and are still leagues ahead of any other cartoon characters in terms of mere richness of personality alone.
Bugs is a few revolutions in one:
The first heckler cartoon I can think of-(if anyone else knows of one before this let me know!) is Porky's Duck Hunt made by Tex Avery in 1937.
Tex Avery loved hunting cartoons-he made a zillion of them. In most of them the befuddled hunter is pitted against many ducks or rabbits of creatures of the wild. In this cartoon Tex' idea was to have Porky outwitted by a flock of ducks. One of his animators, Bob Clampett, suggested they boil it down to 1 duck and make the duck have fun outwitting Porky and be completely unfazed by the threat of the hunter's gun. This gave the animators a chance to spend time seeing what they could do with the new duck character.
(This calm character was inspired by a hunting trip that Rudy Ising once described to the crew at the Looney Tunes factory in the early 30s. He kept acting out scenes of him trying to shoot this rabbit, and the rabbit kept disappearing behind bushes and then sneaking up behind Rudy and completely outwitting him. Clampett and the other cartoonists drew gag cartoons depicting Rudy versus the wacky wabbit.)
Avery's unit tried a couple different takes with the new duck character. Daffy is calm in some of the scenes and he becomes loony at the end of the cartoon in a crazy scene animated by Clampett of the duck spinning and flipping and yelling Woo Woo, Woo Woo!
Shortly after Daffy's Duck Hunt, Bob Clampett was promoted to director of his own unit and took some of the Termite Terrace animators with him-including a young Chuck Jones. This unit carried on the crazy stuff from Tex' unit and took it even further. It also took the personality stuff a lot further while it was at it.
Avery's style was always sarcasm, parody, wacky gags and high concept story premises. Clampett's cartoons were more varied in theme and concentrated more on the performances of the characters. There were really wacky gags too, but they seem even wackier because the characters were richer and the characters, rather than the the director motivated the gags.
In his first cartoon, Get Rich Quick Porky, Clampett experiments with the calmer type heckler character-this time a gopher who looks and acts a lot like the early 40s Bugs Bunny.
He does magic tricks from halfway out of his own hole.
Clampett himself was a magician and a practical joker, and this gopher seems almost to be an animated caricature of the guy I knew well.
The gopher does wacky show off spins into his hole
The triumphant spin into the hole became a staple of Bugs' routine later.
More to come...! The first Bugs Bunny cartoon!
A lot of this historical information I got directly from classic animators but another great resource was Mike Barrier's Funnyworld magazine in the 1970s. It was the first magazine and animation history source to really seriously look at cartoons other than Disney's. Mike and his partner Milt Grey interviewed dozens of old time animators and shared much of their new-found knowledge with rabid cartoon fans like me.
Here's a great interview with Bob Clampett:
Barrier has more interviews on his site: