In order to get this achievement, and in order to play and/or host Co-Op online, you must have at least finish the "Wasteland Thugs" quest first. Since its only a solo quest.
After that, you can play the next quest "The Chosen Ones", which can be done in Co-Op, and can be done 10 times on Easy difficulty. Just play your best character with your stats up and scroll skills up.
You'll most likely get this done quicker with Kenshiro after you finish the Legend Mode with all of his Signature Moves and Skills/Scrolls almost maxed out.
To see how many Co-Op games completed, go to the online menu, select Player Info (completely optional), and it will show you the No. of Co-Op games won.