To earn Stargazer, you must win ten matches of Team Deathmatch on the Observatory map, which is 8v8.
This can be earnt as soon as you have finished the tutorial.
To enter PvP, press right on the dpad. Press right on matchmaking, then right on observatory and select enter match.
You will then queue up, and when the queue is finished you will enter.
Simply win ten of these matches, and your achievement will pop. These are team wins, so as long as your team wins you are golden.