Finish all 24 Races in 3rd or better. Tracks 8, 16, and 24 are unique course designs.
Make sure your car is fully powered up before you get to the last race, as you'll need all the boost you can get. The blue car is very quick here. Quick passage through the last turns will really help.
Hint: If you can, avoid using too many weapons - the yellow car will take off and lap you. If you must use weapons to maintain your lead, make sure they're on the mark. Rewind is very helpful here. I still had 75 bombs at when the achievement popped.
(FWIW, I could never finish this track on the NES original.)