Why does Damon want to kill Elijah right now?

Because Elijah's deal involves Elena dying

I think it is because he doesn't trust Elijah that much. He doesn't believe Elijah will keep his end of the deal.
Also the fact that Elijah is putting Elena's life in danger might have a part of that :)

I think that Damon want's Elijah dead, because deep down, he loves Elena. And doesn't want her to be sacrificed, which means she has to die. And he will do anything to protect her.
Also, i dont think he trust's Elijah, and probably think's he will break his half of the deal. :)