For a description on how to gain Overkill and what it means, refer to the description for 'Overkiller', above.
To unlock this achievement, you need to kill 200 enemies while in Overkill that your partner has triggered. If you are playing with a human partner, have them press when they have Overkill ready to activate and then proceed to kill as many enemies as you can. If you are playing with an AI partner, you can let them activate it on their own, or you can activate it yourself by pressing on the d-pad. You will know when it is ready to activate, as the map of the d-pad in the bottom left of the screen, will have the :down: button flashing orange. Repeat either of these until the achievement unlocks.
These kills are also cumulative.