The tutorial teaches you how to do a handbrake 180s, all you have to do is quickly swipe downwards on the screen whilst turning sharply in either direction.
The achievement can only be won as a racer, not as a cop. I just loaded up the first race and continuously did 180s. You have to get up a bit of speed, i found above 85-90mph was sufficient in order to execute the full 180.
Once you've done 10, continue to the finish line (you will be last, obviously), and the achievement pops as soon as the race is over.