For this you will need to complete the first act of the game with each character: Logan, Sam B, Purna and Xian Mei. This can be done in any order, and only the main story quests need to be completed.
Note: The achievement tracker in-game appears to be broken, as it will sometimes not remove anyone's name even though you've finished the act with them, or remove the wrong person's name. The achievement still works though. If you don't get credit for completing the Act with someone, start a new file with them at Chapter 4 (this can only be done if you've completed the chapter previously) and then finish it again.
This CAN NOT be unlocked by joining someone else's co-op game as they finish Chapter 4. You do need to complete the entire act's main story quests, not just "finish" the act. Just run from location to location doing what you need and it will go rather quickly though. Avoid any fights that aren't required to progress.