To get all of the rewards in a Co-op mission, all you have to do is replay the specific mission until you unlock all the articles of clothing that show up under the description for the mission (the boxes will be checked off after you attain each piece of clothing).
The easiest and quickest mission for me (and my co-op buddy) was the mission "Heads Will Roll" (it has a 2 out of 5 rating on difficulty and is pretty simple). You can play this mission by yourself and it's not required for you to be playing with another person. You only have to collect three articles of clothing for this mission so that means you only have to complete it three times. You don't have to do it in a specific way (such as sneaking through the mission, etc.), so all you have to do is just beat the mission three times and the achievement will be yours.
I hope this helped, and have a nice day!