This achievement along with No-Nitrous is very time consuming. There are 30 single races. You need to win all 30 races with all 5 car classes AND without getting into a collision(150 races total).
Tip: Turn on unlimited nitrous(which is done by hitting Y as you're selecting your car.
Don't be afraid to restart the race often, the cars won't change (as long as you don't exit to menu). You will begin to see a pattern in their driving habits.
Once you're past everyone else, stay in the middle of the track. This lessens the possibilities of hitting a wall or rail.