Originally Posted by Paste Pot Pete
Actually, he obviously tried to kill him a couple times.
He blew up Axis Chemicals (and every goon inside), and you can assume he thought Joker was inside too.
But if that's not enough for you, he aims his Batwing machine guns and missiles DIRECTLY at the Joker.
Then, confronting him in the tower, he says "I'm going to kill you" - before punching him right off the rooftop. Had Joker not held on, he'd have been dead right then.
Killing Joker is the best thing Batman could have done. That way he's saving a number of future victims. In Burton's movies, Gotham is so corrupt that jail might not be an option; Grissom was a dangerous mobster and a killer and he was always a free man.