Lex doesn't actually want to kill Batman - he wants to kill Superman.
All Lex Luthor wants to do throughout Batman V Superman: Dawn of Justice is kill Superman, as he fears that a man as powerful as Superman will one day become a dictator of sorts, using his godlike powers to take over the world.
By forcing Superman to fight Batman by taking Martha Kent hostage, Lex hopes that Batman will use the Kryptonite he stole from Lex to kill Superman - which very nearly happens.
It's worth noting that Lex has been baiting the two of them into battle for some time, setting up a series of events that lead Bruce Wayne/Batman to come to the same conclusion that Lex has - that Superman is too dangerous to live.
As such, Batman is nothing more than a pawn in the game that Lex Luthor is playing.