The best way that I've found to do this, and any other "Cops & Robbers" related achievements, is to play on Area 41 in the Desert region with Aston's BullDog. It's a very fast car that has great acceleration and handles drifting perfectly.
You're going to want to do your donut as early in the race as possible so you can focus on getting your checkpoints faster which will get you more money and get you your win. To do a donut, hold
while holding left on your
. Now after you complete your donut, just finish the race as quickly as possible and you should get first place which will net you the achievement.
I've had multiple friends attempt these achievements under the same guidelines and they had no issues. Let me know if you have any other questions.