I mentioned this in my extensive guide for the CarBoom! achievement but I think this achievement is currently glitched in a way beneficial to us achievement hunters.
My in-game achievement tracker says I only have done four of the achievements requirements but I've already unlocked the achievement on my gamertag.
My guess is that you're supposed to destroy cars with human players in them but for whatever reason the game gives you progress outside of the game for blowing up empty or AI controlled cars. I'm not entirely sure the criteria for gaining progress but if you shoot at cars or throw grenades while doing missions in an attempt to win then you should have no problem getting this.