When you are at your Club House, going to the bar and creating a race, "Inner Club Race" will allow for you to drive toward the 150km/93 miles. Any race within the Inner Club race will count towards it and it goes by your own mileage, not the race-type. So the Keep Your Distance mileage is not what it goes by.
The best race type to do this on would be "Follow the Leader". Having either 4 or 8 players is necessary. Just simply drive in the opposite direction of the checkpoint, making sure you don't go through it and keep driving until the achievement unlocks is probably the quickest way to do this.