Some liberties had to be taken to pull this off, but this still is the best you're going to do at a shot-for-shot recreation of The Fresh Prince of Bel-Air's introduction, inside Grand Theft Auto 5 anyway.
I mean, that's it. It's right there in the lede paragraph. Watch for yourself. Franklin stars as Will Smith and Rockford Hills subs for Bel Air. There's some firearms play substituting for other things that couldn't be reproduced. The cop at the beginning also beats Franklin upside the head and the fellas who are up to no good attack him with a hatchet on the basketball court. That may have altered the original premise somewhat.
If you want the side-by-side comparison, it's below. This is the work of YouTuber Merfish, the guy who did the introduction of Full House in GTA 5 earlier this month, and he promises new GTA 5 videos every week. Clearly, the next one he should do is Blossom.