As of patch 1.4 (15/12/2015) this achievement is still bugged and doesn't track the progress properly, it has been reported multiple times on Steam Forums.
I found out that you can actually start up solo lobbies and do acheivements that do not require kills due to you being completely alone.
At main menu click "Create a Game" then choose the map you desire, so for this achievement it is "Toybox Assault".
Then down the bottom of your screen, Tick the "invite Only" button and press "P2P" this will take you to a lobby and you can just press "Force Start Timer" and away you go :)
For this, go to the track in the middle of the map as it is shorter so the time it takes for the RC Cars to circle back round and hit you after you respawn is relatively short. Just let the cars hit you repeatedly for 20 times until you unlock the achievement. (However look at the top of the solution it is currently bugged and doesn't track properly)