A new addition to the Saints Row game is the inclusion of the Co-Op "competitive" mode. This is a very simple mode that once turned on, it will stay in effect during any Story Mission or Strongholds mission. It basicly pits you and your buddy against each other in a scoring mini-game each mission. The person who does the most property damage or get the most enemy kills, wins.
Each thing you destroy give you a small value of points. Each enemy you kill will give you more points than that of property damage and every enemy vehicle you kill will net you even more points. Simple, yes, but adds a great edge to the already fun co-op.
You turn this mode on under the "Co-Op" tab from the main menu while in game. You have to turn it on BEFORE you and your partner start a mission. Turning it on mid way through a mission will just tell you that it won't start till the next mission.
For the achievement to unlock, just win this Co-op competitive mini-game 5 times. If you have already beaten the story and all the missions, the easiest way to replay levels and get wins is via the Notice Board in any of the Safe Houses. I found the Stronghold were very quick to replay and win.
If you want to get this out of the way, just take it in turn to win the game by having each player not do anything while the other gets all the kills.