The game can stick a little on tracking this.
You need to complete 5 rounds on a handicap rated course for the game to work out your handicap. In REALLY simple terms, this is an average of your shots over or under par for the course.
To get Scratch, you'll need to average a Par score or better over five rounds of golf. Louisiana and Summerlin were my go-to choices.
You can monitor your handicap rating and courses played in the start menu.
*TRACKING FIX* I've seen that the game doesn't always immediately record the rounds played for your handicap score. I had completed 2 rounds, then shut down. Completed 5 the next day and when I checked the handicap rating, could only see the 2 rounds from the day before. If this happens you'll need to shut the game down and restart. The game will log the rounds and work out your handicap rating. If it's below 0 at that point, you'll need to play one more round to trigger the trophy. (You can play a half round local game and it will still trigger)