To gain access to Twinsen's car, on Desert Island, head to Jerome Baldino's house, where Baldino will give you a component for your unfinished car. Head back to Citadel Island by ferry and give Zoé the car part. She will then tell you to return to Baldino who means to give you a portable radio.
Once you speak to Baldino and receive the radio, Zoé will contact you and tell you that she has fixed the car and has brought it to Desert Island. You will find her and the car parked beside where the ferry is docked.
At the race track (highlighted on the above map), drive to the starting line and wait for the race car driver to pull up and stop. I recommend creating a save file just before the driver pulls up because this may take a few attempts:
The driver will rev three times before setting off. Try hitting the gas at the same time as he does (without overtaking) to keep up with him. If he says "Lost!!" anytime during the "race", you will need to restart - this will happen if you either stray too far behind, drive ahead or collide with him.
If the driver drops a clover at the finish line, the achievement should also pop.
Once you leave Twinsun for the Emerald Moon, you will no longer be able to get this achievement.