Here's how to unlock this trophy eezy-peezy:
Go to Chapter Select and pick Chapter 4 - The Dig, and load it up on Very Easy difficulty. Progress a very short ways into the level until you get to the shed where you get the grenade tutorial, next to the boxes of grenades. After the cutscene, skip through the tutorial and run out toward the guys coming at you. About 20 to 25 enemies will spawn here, so go to town on them until you've killed 20 of them with the "square" melee attack.
Don't worry if you run out of enemies before getting it. Just reload the checkpoint and try it again.
While you are doing this, you should be working on the following trophies as well, so see my other solutions for them while you're at it:Note: The trophy will unlock as soon as you get the 20th kill in this manner.