You must complete all 5 acts (all 31 chapters) in 4 player coop. It does not need to be all at the same time or with the same people. It does not need to be the same difficulty. It can be campaign, arcade, or a mix of the two. It does not need to be in order.
I had some people lag out during the game. We just paused the game, invited a new player into the game and continued the story without having to go to the main screen.
I would recommend to do this in arcade. It counts the same as campaign progress, it contributes to online ranks and ribbons and most importantly, there are respawns: campaign makes you restart the checkpoint whenever someone dies. If you haven't gotten it already, unlocking this achievement will also unlock
If you have any questions please ask away in the comments and I will update accordingly. If you downvote, please tell me why so I may fix the problem/inaccuracy.