On the bottom floor there is a display case with a rocket launcher in it, get it, and shoot him while the lights are off and facing somewhere else, he will hold the rocket and while he does that you need to shoot it, after that run up to him and hold himdown to inject him.
If you miss the shot, on the top floor you'll find more rocket launcher ammo.
This is what I did:
To avoid missing the rocket launcher shots (Wesker dodges a lot of them), and then having to go find and get another rocket, I decided at the beginning of the fight to acquire all items from the breakable crates, then collect all rockets and put them near one of the "light stations" you have to turn off, get the rocket launcher itself and then turn off the light where i left my rockets in a huge pile of like 10.
This allows me to reload faster without having to go search for another rocket every time I miss and also stay around the light station you've turned off, that way you'll have a better chance of hitting Wesker. Don't worry about the other lights.
And as everyone else said, once you get a good shot in and he's stopping the rocket from hitting his face, you explode it in front of him and the rest is just being able to follow the screen commands.
Hope this helps