Originally Posted by DaReAlT83
Only destroying by crashing counts. The only mission I remember where to do this is "Guests of the State". Unfortunately, only the last of the cars you have to wipe out has to be destroyed that it counts. So you have to play the whole mission over and over until the achievement pops up.
Yesterday with the help of another user I figured it out. The easiest mission for this (for me) was on Day 4, called "The Party". In part 2 of this mission, you get to crash a party of some drivers and need to follow one by destroying him.
I picked the fastest car available, got up to speed and crashed into him into the rear. After doing this about 5 to 6 times, the car was successfully crashed and counts as a kill.
Thanks for mentioning about the last car about the other mission, though - I tried it for the first cars and always failed :-)