Ultimately I agree with Ly. Someone else would be the one to kill Joker and then probably be pursued by Bats.
If Batman was the one to do the deed though I could see it going down a number of ways. The one that makes the most sense to me would be something like in DK Returns,
. Also the act of killing the Joker would have a tremendous effect on Batman's already fractured Psyche. He's basically already insane. In fact, in order to make the actual decision to go out and finish off the Joker would be something of a breakdown.
Then let's say he does kill the Joker, completely willingly and somehow comes out mentally unscathed. Ok well not entirely unscathed cause he is about to blow up Arkham. He could keep it under wraps, and when the place goes up, for all the public knows, Batman didn't even do it. Sure he killed the Joker, but that was the Joker. Unless he tells Gordon beforehand, then they would probably look to Batman to catch the person whom they suspect did it. Hell, Batman would probably even turn himself in. At that point, alot of danger is taken care of, and if need be he'd most likely find a way to break out of jail. In comic land anyway :p lol