This will likely be the most difficult achievement in the game. If either player dies it's game over. However only one player needs to reach the exit to complete the stage, if the other player dies or times out after the other player has reached the exit the stage will count as complete. Completing the game on Co-op normal then moving onto insane with the same partner if possible would be a good way to get familiar with the Co-op mechanics and levels. Good communication would also be helpful for completing these as painlessly as possible. Use the Achievement Trading Thread to find people to play with online.
As in Solo Insane Mode, the Acid Bubble is extremely useful and will often allow one player to traverse obstacles that would otherwise require a co-op action. In some places however throws, grabs and up attack pushes will be necessary to proceed. Jumping on top of the other player's will act as a platform, this will work when they are climbing or in the air. As only one player needs to reach the exit, some levels can be made much easier by having one player stay behind at a switch or an inactivated exit.
Co-op mode has some additional challenges related to the design of the game. In both online and local play, the screen will zoom out as the players get further apart, making it harder to see. However you will often not want to stay close to your partner, as players impacting each other will very commonly lead to death. In online play exclusively there are problems with lag. It will often be lethal for the joining player to pick up rocks, so plan to have the host do all the heavy lifting. Sometimes lag can get more severe to the point where co-op actions won't work properly, the joining player will fall through lifts, and weapon-tools will activate late to the point of being useless. If this starts happening it could be a good time to take a break or reset your router, nothing is more frustrating than a good run ruined by the game glitching out.