I found this one to be a bit of a pain as if you hit the car while trying to get close to it this will void the achievement for that car. The best method that got me it quickly was to go onto chapter 1-1. At the start of the level you will get passed by a single car, now try and boost past the car. If you are too close to it you will ram it so just zoom past it then keep on hitting Y. With any luck he'll be right behind you. This was the last achievement I unlocked so I was fully upgraded and the car went down in one hit.
Edit - This is a video guide for this achievement. Thanks to LSC MrSandman for providing this one also.