For this achievement, you must first join or create a Syndicate. You can then start contracts as certain players will excel in some categories while you will find certain aspects of gameplay that you are better at. These contracts begin to occur when other members of your syndicate advance further than you in any challenge and can include kills, reboots, heals, saviors, executions and variations on most of these.
To fulfill a contract, you need to surpass and essentially beat a syndicate member that is higher than you in that category. If you are not as advanced in one of the many kill-based categories, you focus on other categories such as heal- or reboot-based challenges. There are essentially many categories that allow players of different strengths to be competitive in their syndicate and I think these contracts are motivational tools for you to keep your edge while playing coop. The achievement is much easier if members of your syndicate are near you in rank and challenge progression because the numbers needed to complete contracts will be lower for you if you can keep up with your buddies.
I could not find an actual in-game counter for the total number of contracts completed, but you will see them appear in the post-game screen after completing a coop map. The completed contracts will be listed with your other accolades such as blueprint tokens acquired, challenges completed, chip upgrades acquired, etc. If you are not getting this achievement, it may be due to the fact that you are "top-dog" in your syndicate and you may have to allow your other syndicate members to catch up and pass you so that you can once again climb back up to the top and complete those contracts along your way! Happy hunting!