For a description on how to gain Overkill and what it means, refer to the description for 'Overkiller', above.
Double Overkill, is when you and your partner both activate Overkill at the same time. You will know when this has happened, as it will appear on-screen informing you. To perform a double Overkill, wait until you both have the ability ready to use then activate it at the same time. If you are playing with a human partner, you both need to press at the same time to activate double Overkill, but if you are playing with an AI partner, you need to press for your own Overkill and for your partners overkill.
After you have killed 200 enemies in double Overkill, this achievement will unlock. The kills, again, are cumulative.