To gain Overkill, you must kill enemies, until your Overkill meter on the left side of the screen fills up. When it is full, press to activate it and you will become invincible for a short time, your bullets will become explosive and you won't need to reload. You will gain more Aggro towards your Overkill meter by using loader and more powerful guns, like the shotgun. You must kill 200 enemies while in Overkill, that you have triggered, to unlock this achievement. Kills during Overkill that your partner has triggered, whether human or AI, will not count towards this.
The 200 kills are cumulative, so you don't need to kill 200 in a single Overkill. Single Overkill just means an Overkill that has been triggered by a singular person, as opposed to a double Overkill, explained in the description for 'Double Trouble'.