You can participate in bike races after you complete the first couple missions in the story. You can only do this in single player, multiplayer bike races don't count.
To knock someone off their bike, press if they are on your left or press if they are on your right. Hold down the button for a couple seconds to hit them harder. If you hold it down, they will fall off the bike once you hit them but if you just tap it then they won't fall off.
There are 12 bike races total and you can replay any race as many times as you want as long as you don't win. I suggest on the first bike race, whack as many people off as you can. If you don't win, wait a couple seconds and you can redo the race. The race ends as soon as the first person crosses the finish line. The easiest places to hit people are right at the start because everyone is close together, and if someone is in front of you and you are gaining on them, you can easily pull up right beside them and knock them off. If you come from behind, make sure you hold the button down so you can knock them off right away.
If you don't whack 69 people off their bikes by the end of the story, you can still go back and race again.