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?