There are a total of 12 rivals for you to beat. Most of them require 28 "Rival Points" to beat them, with the final rival, cover athlete Justin Thomas, requiring 35. You can get up to 7 points per round based on:
-Front 9 Score
-Back 9 Score
-Birdies or Better
-No Bogey Round
If you get all 7 points every round, it will therefore take 4 rounds to beat most rivals and move on to the next one. If they tie you or beat you on any of the criteria, it may take an extra round to get the points you need to win the rivalry.
You can view your status by going to the PGA TOUR Career Home menu and using
to get over to the Records tab. From there, select My Rivals. It will take you about a season and a half total to get through them all so just keep playing in every tournament beating them on the 7 criteria as best you can and eventually you will beat the last rival. You can play with the season difficulty on Very Easy and your golfer difficulty on Beginner to make it go faster.