The only way to defeat him is to inject him with the serum after stunning him with a "specialized" rocket launcher or with a powerful weapon (such as a fully upgraded magnum pistol) in Chapter 6-3 - Bridge Deck.
After you do that, a long cutscene will play. (It will involve several moments where you must press a button to avoid getting killed by Wesker)
When that cutscene finishes you will end up in the summit of a volcano, where the final battle against Wesker will take place.
As for how many bullets to kill him at this point... well that depends on what kind of weapons you've got at Chris' and/or Sheva's arsenal. I don't really know how many. If your desperate, you can try counting the gun-shots if you like, the next time you play the game.