When playing the PGA Tour campaign, you will be given a rival automatically (I was given Patrick Cantlay on my second tournament). You then have to beat them in 7 categories, such as lower score, lowest score on a hole, etc. You don’t have to win all in one go, as if you don’t, the rivalry will carry over to the next tournament. When you have won 7 rivalry “points” you will beat them and this achievement will unlock.
After every round, it will update you on where you are against your rival.
As such, you should get this over time.