The thing that bothered me most about the 89 Burton Batman was that the Joker died. If you grasp the 50 plus year history of the characters you know that when Joker dies, so does Batman because they are each other's direct opposite. So, I'm glad he didn't bite it in this film. For Batman to go on, the Joker needs too also in some way.
Two-Face was fairly accurate, though in the comic he was more of a playboy than what we saw in the film and his motivations didn't involve the death of his girlfriend. Originally, he was just another throwaway villian and he really didn't come into his own until the mid 80's, though he first appeared back in the 50's. In the comics he was scarred by acid. I think in Dark Knight they gave him alot more depth than the comics ever did.
Batman 101