You can play Contracts by pressing L1 or R1 from the main menu of Hitman. Once selected, you have the option of playing "Featured Contracts," "My Contracts," or "Latest Contracts."
Featured Contracts are Contract Missions that are showcased by the developer.
My Contracts are Contract Missions that you have created.
Latest Contracts are Contract Missions that other users have created recently.
Each category counts towards the trophy.
A Contract counts towards completion when you kill the mission-specific target and escape. Select "Exit to Main Menu" at the mission complete screen. If you fulfill additional mission requirements during the contract mission (Optional Objectives like wearing a specific outfit or killing the target with a specific weapon), you will have a significant score increase when you successfully escape the mission area.
Keep in mind a completed Contract doesn't count towards the trophy score until you choose to "Exit to the Main Menu" at the mission complete screen. From my experience, "replayed" Contract missions do not count towards the 10 needed for trophy unlock.
*If you have replayed Contracts and had it count towards the trophy, please let me know and I will edit the solution*
I've found the fastest way to complete Contracts is to select from the Latest Contracts and search for simple one target contracts people have created. Mostly they've made it for the "Top of the Class" trophy but it's also a good opportunity for you to get the trophy too.
There's no way to keep track of how many Contracts you've completed from the game menu. If you'd like, you may want to keep track of how many you've completed so you're not guessing but it is only 10 so it's up to you.