Why does Kaido want to kill Whitebeard? According to Wikia, Kaido's article:


  [...] Kaido and Edward Newgate were not allies, yet would not, under normal circumstances, provoke each other. However, Kaido saw the opportunity to take down his rival prior to the impending Battle of Marineford, and did not hesitate to take advantage to attempt so.


I don't know why Kaido wants to kill Whitebeard at first place. Are there any evidences from the series that suggest so?
Pirates are pirates. If they see a chance, they will take it. Also Kaido could have gained much with Whitebeard's death.

According to the Wikia article about the Yonkou (sorry, I couldn't find the chapter where they actually say this):


  [...] the Yonko stuck in a deadlock situation [...]


Ideally Whitebeard's death would have given Kaido the advantage he needed to advance in the New World and eventually become the Pirate King. He just didn't expect Shanks to help out Whitebeard, or Blackbeard to be so swift taking his place and becoming a Yonkou maintaining their deadlock.