Credit to AceCardJones from TrueAchievements for this guide:
This is easiest to do on the Egg Hangar stage in the Classic Cup. I used Sonic, standard mod on B class. I went B class as I found the AI stagger more quickly on C class as the rear of the pack are really slow. after the long straight with boosts, items and robots patrolling the track, you will enter a really really long left turn that you can drift in it's entirety, if you go from there in 10th or 9th and over take as many as you can you should easily manage it.
Even as that long left turn ends it then switches to a right turn which if done correctly and quickly enough you can switch to a right turn drift without interrupting your initial drift if you only take your finger off drift for a split second and start again instantly so you have ample time for this achievement provided you don't crash or get hit. Restarting this a few times at worst should yield the achievement. Pops immediately.