You can get this achievement in both the single player mode (with a co-op partner or computer) or in multi player mode.
I would imagine it is best to do this in the multi player mode however I got this in a single player co-op match.
You will unlock this three different ways. If you play through the game twice/three time (to beat professional and easy mode, and maybe to get the My Virtual Friend achievement), if you boost the 1 billion dollar achievement (one guy with us got both the overkill achievements at around the 700 million mark), or boosting for it.
To get power mode overkill you will want all the aggro. If your playing with a AI partner put him on passive mode and max out your aggro meter (you will need to kill at least one person to max it out). Once it is maxed out you will need to fill up your dark red meter by continual gun fire when the dark red meter inside your red meter is full for 2 seconds or so, hit the A button when the overkill message comes up to go into Overkill mode. You will then need to kill 100 enemies to unlock this achievement.
If your planning on boosting for it either do it in multi player mode or in story mode right after a checkout with 7+ enemies in the room. You can continue to reload the last checkpoint to get this achievement (this is how I got it).