Jason Todd is a vigilante anti-hero who has been a member of and antagonist to the Batman Family. Batman originally trained him to be the second Robin, his new side-kick after d**k Grayson became Nightwing. Jason is murdered by the Joker during A Death in the Family, although he is later resurrected and returns as the second Red Hood during Under the Hood.
His portrayal has at times been inconsistent. I myself would prefer him to be an anti-hero who walks on the edge of the abyss, I think DC needs more interesting characters like that, and Todd's story would be fitting to that end. However he has at other times been portrayed as genuinely insane or psychotic. He even tried to kill d**k and Tim, something I think shouldn't really be in character for him. I see him as a guy that ultimately wants to rid the world of bad guys and isn't afraid of killing to accomplish this. Hence, he doesn't really have a reason to try and murder Drake or Grayson. Besides, it's completely off character for him: prior to Battle for the Cowl, he had already beaten Tim Drake to a pulp and didn't saw fit to kill him then, Why would he suddenly decide he deserves to die later on? He also shot Damian in that arc (which was awesome), and trying to kill a child doesn't seem like something Jason would do. Then again he might have been counting on Damian's body armor to save him. I think Jason would obviously resent Bruce and the bat-family, but would still respect them at some level while trying to prove he can be better than them.
Jason's code is like that of an anti hero, he isn't afraid to kill scumbags who really deserve it, but wouldn't kill innocent bystanders or guys that actually try to make the world a better place.
But then again, maybe I'm just talking about how I'd like to see Jason develop as a character instead of how he's been portrayed since the "red head-silly costume" change.