Once you slide down the rope after defeating the 1st (and only boss), you will need to kill ALL enemies, without letting any die from explosions. With your flail gun kill the 2 enemies in front of you, and progress through the level. You will then need to kill the turret guy on the walkway, and the green scalper that jumps from above. Next the door to your right will open, revealing 3 enemies. These enemies are the first enemies vulnerable to an explosion. Make sure to kill all 3 before the explosion can, and progress to the walkway where the turret guy was. Take the turret and turn around 180 degrees. Start on the right side, and kill the 3 enemies, now aim to the left side and take out the 3-5 enemies. After taking them down, it will trigger the explosion. If you don't do it fast enough, and one of them are alive in an explosion zone, then you will not get the achievement. Keep aiming on the left, killing the enemies that run out, and then back to the right hand side, killing the last 3 enemies, triggering another explosion. If any enemy is alive during this last explosion, you will not get the achievement. If you have successfully killed all enemies, the trophy will unlock during this moment.
For more info, please reference this video walkthrough: