Personality
Lester may be extremely skilled and deadly, but he has two great limitations. The first is that he seems to always be drawn to foes even more skilled than he is. The second is that he's clearly unstable.
Lester is a sadistic (former) killer with a strong macho self-image. His fighting ability and number of victories are very important to him, giving him the self-delusion that he is strong while he's, in fact, quite unstable. Given this delusion, it is unsurprising that he experiences great rage with those who best him. Obsessively so, in many instances.
Defeating such an opponent can fill him with great pleasure, and he's been known to take trophies from particularly challenging targets. While quite intelligent, his obsessions have worn him down and his actions sometimes do not make much sense, often tripping him up at the worst moments.
Oddly, he does have something resembling a moral code, as he has never actively hurt or killed a child or animal. He refuses. He may not have lostt much sleep over them as collateral damage in the past, but he refuses to harm either, possibly a result of his potentially abusive childhood.
His penchant for heroism is new and surprising, even to him, and he sometimes struggles with the concept of the Right Thing. His world was black and white for so long that shades of gray baffle him (ironic, given his color-blindness).
As far as personal relationships go, he's too broken for anything normal, and has generally eschewed long-term relationships. Unable or unwilling to share even a tiny truth about his past, he is generally unable to make strong connections, tending more towards antagonism and derision than honest communication. He learned a long time ago that the only one sticking around for the long haul was the guy in the mirror, and -he- was a dick. Maybe he can be something better, now.
Background
It depends on Lester's mood as to which history or name he'll give you. For example:
The first person Benjamin Poindexter killed was his mother. He shot her in the head after she proved inadequate defense against his abusive father.
The first person Leonard McClain killed was his father. He painted a bullseye on his father's forehead as he lay passed out from yet another alcoholic bender, and shot him point-blank in the center of it.
The first person Lester Jangles killed was a minor league baseball player. He killed him with a 105mph fastball to the head.
The first person the assassin known as Bullseye killed was whichever one of those was actually him. He liked the stories - and the name Lester, so he kept those.
Whether any of that is true is debatable. What is true is that he had innate ability to hit a target with deadly accuracy. So he trained. And he honed himself. He learned from acrobats and martial artists and master assassins, some of whom he killed.
His skill got the attention of the government and the Weapon X program. They made him a better killer. They gave him indestructible bones, and access to all kinds of military ordinance that was good for killing a lot of people. Sometimes all at once.
When he got away from the government, he killed for money. Criminal overlords and shadowy villain-types with a lot of money and an axe to grind. They made him comfortable, and weren't fussy about collateral damage.
Eventually, the killing had to stop. His entire ledger was written in red, and the government was after him again. It was then that Osborn's men approached him and offered him an out: stop killing and go legitimate, and they would get the government off his back. So, he tried being a hero. It didn't seem all that hard, and he was better than most of the heroes he'd seen.
And, the damned thing about THAT was that being a hero was way more interesting and rewarding (well, not financially) than being a killer ever was. He may have started out on a leash, but in time, he found himself less likely to kill and more likely to dispense justice. He still liked kicking ass - a lot. A -lot-. Only now he's kicking the ass of his former cohorts.
Life's funny that way.