As we all know that the Joker abducted Jason and killed him in a dungeon. Before he killed him he convienced him to hate the Batman, he tortured him and made him understand that the Batman is responsible to this situation which Jason has gotten into. He died when the Joker blowed up the whole place.
After Jason's death he has been ressurected by Ra's al Ghul. He put Jason in the Lazarus Pit and raised him from the dead.
Jason may hate Bruce but he hates the Joker too, so he wouldn't become what he hates, he wants to kill him and get Justice for the things he's done to him. Also, Jason isn't a villain, he's an anti-hero who thinks Batman has a weakness, he can't kill! Jason became the Red Hood to clean up the mess of Gotham by killing the people responsible, including Joker and especially Batman because he thinks he's the one who put him in the Joker's crosshairs. That's why he's not the Joker in Batman vs Superman.