Really easy achievement.
Press left on the D pad and that will bring up the scoring screen so you can see what your score is for each corner.
Then when you come to the corners just hold X button or A button and that will make you drift but at same time using the right trigger to keep accelerator, but you must finish the drift, so you arent allowed to spin the car out or go off the track or it wont count the points.
And once you pull off a drift over 1000 points your achievement will pop up!