This achievement will be a time consuming one to boost so really just try to play the game normally and it will come naturally. It is a cumulative achievement so it will add up over every game match you play.
An assist is where you aide another person in taking out an opponent, but you don't deal the final blow. Just keep shooting enemies in online/offline multiplayer matches and if you deal damage but don't deal the final blow you will see the word Assist.
You can check via the paused achievements list how many you have accumulated.