I'm not at the part in the mango where Law teams up with the Straw Hats, and I don't recall if he mentioned what his goal was. Does he also want to be the king of the seas?
His most recent motivation was a long term plan to take revenge on Doflamingo by taking him down. I'm guessing he wants to take out Kaido too because of his relation to Doflmaingo. They probably had similar plans in play.
So you don't think he's got any ambition outside of that? Like what do you think he would do if Luffy was about to become Pirate King? Would he challenge him for it?
I think at this point Law has decided to throw in with Luffy to take down Kaido. Though his plan originally was never to survive Dressrosa and die taking down Doflamingo.
After Kaido is defeated he'll probably be part of Luffy's new alliance like everyone else.