Get familiar with a level you've played on, and play it on Time Trial. When doing your race, make it a point to avoid the cars, even if it mean swerving into a wall if necessary. On your map, there are grey dots indicating when traffic is coming up - use this! Also, use your map so you know when to use your nitros, and to pay attention to sharp turns coming up. Sometimes, if you get close to a car, it is hard to tell if you hit them or not - if you don't get the achievement at the end of the race, then you most likely accidently hit someone.