As the description says, just complete episode 1 on "Hurt Me Plenty" difficulty or higher. While "Hurt Me Plenty" may be the easier option, completing it on "UltraViolent" will help towards the Rampage achievement, and will save you having to play through the episode twice. Please note this achievement cannot be obtained in co-op, it has to be done in single player.
If you do decide to play through it on "UltraViolent", remember that saving the game often can help, especially if you don't know what you're doing. It really isn't overly difficult, as long as you remember to conserve your ammo, and make sure you take your time and don't rush head first into fights.
Thanks to DaveOwl, I can confirm that you don't have to complete the secret level (Military Base) to get this achievement. And thanks to deutschZuid, I can also confirm that you don't need to complete this level to unlock the Rampage achievement. If you still wish to complete it however, it can be accessed from the secret exit in E1M3, which can be found relatively easily with a bit of exploring. If you still can't find it, just check the YouTube video to see where it is.
In the final level in the episode, Phobos Anomaly, don't be afraid to break out the big guns, as you lose them after this level anyway. The Rocket Launcher is probably your best bet, as it can take down Spectres in one hit, and the Barons of Hell go down after about 5 each. Once you beat them, the way to the exit opens up. After you step through the teleporter, you appear in a room where you are attacked by a bunch of enemies. You are supposed to die at this point, so let yourself get killed to end the level and the episode, and you'll get your achievement.
If you have any trouble with any of the levels, just click the link below to find layouts of the levels, as well as their secrets, and tips on how to complete them. If you're still having any trouble, feel free to leave a comment below or send me a message and I'll be happy to help you out as best I can.
http://doom.wikia.com/wiki/Knee-Deep_in_the_Dead