To get a holeshot you need to be the first rider through the red and white arch setup near the beginning of each race. To better your odds of doing this you can gain a small speed boost at the beginning of each race by holding in your clutch (LB by default) and holding in the gas at the same time. This will fill the meter at the bottom right of your screen showing the potential boost to be gained. The goal is to release your clutch at the same time the gate drops to obtain your boost.
The easiest place to do this is on the first Nationals track with the difficulty set to easy. This track gives you plenty of time to get to the front of the pack and gain your holeshot. A message will appear in the bottom left corner of the screen confirming that you have earned the MotoSport Holeshot. Now just go on to win the race and the achievement is yours!
This is boostable online in a private match if you are having trouble completing it in the single player.