Take note that the berserk powerup lasts the entire level, not just until the red filter goes away.
Easiest way to get this achievement is perhaps on DOOM II: No Rest For The Living, Map 1. Hurt Me Plenty difficulty will be sufficient.
As soon as you spawn in, you will see a grey UAC wall panel directly in front of you. Interact with it to open it and reveal the berserk powerup. Simply pick it up, and then stick to your fists, punching the various zombies and imps that the level has to offer.
Here is a video if you prefer to see it demonstrated :