Need for Speed: Hot Pursuit

Hit up "Mission Beach" should be the SCPD bottom part of the map. Select the first event "Escape to the Beach" pick Subaru Impreza WRX STI should be the first cop car in the lot.

Now they're three racers driving, it's easy to hit them but just try not to... if you do oh well just focus more on drifting the turns. Holding on to your boost helps too so use it before you bank a turn. (Use your map on your radar it should tell you every turn before you hit it) Tap LT (Left Trigger) while moving with the turn you should be drifting... it tells you how much yards you drifted during that turn, sorry it doesn't really tell you how much you actually accumulate but I'm sure if you are drifting every turn in this event you should get it. If you crash it's okay just don't let the racer(s) pass you completely off course. Just make it to the end and the achievement should pop up.

I'm not sure if you get the achievement if you bust all the cars up after you think you accumulate 2,000 yards of drifting.

I follow Miss Jaded's solution but there's one way to finish it quicker. When the race she mentioned starts the road turns right and exactly there is a building (like motel or drive in) you can drift around till you think you got it. The racers almost wait for you, so there's no problem to finish the race after you did a few rounds around that building.
After you finished the achievement pops up. Worked for me.

Alternatively, what you can do (as this is what I did):
I knew I didn't have 2000yrds of drift, so (on a 6.7 mile track) drove back to the start whilst drifting, turned around and drifted back...

The chievo popped after the race finished (yes you finish last, so do it after you've already got first place, or before and win after..)

Lightning reflex, SCPD on the east side of the map.

Suburu Impreza, no shortcuts. Just drift the turns wide and as far as possible, and never crash out of a drift, it doesn't count.

It's easy to get the achievo, just make sure your spotting all the headlights/backlights on other vehicles to avoid them.

On the widest turns, use nitrous too max out the distance on the exit straight. The car will continue sideways for a good distance after the actual turn ended, although this can require practice.

You will unlock this car early on while playing as the Enforcer and the Racer. After you have the car, start an event and drift on every turn. You can also pick an event on Builder Desert such as "Lightning Reflex" for the Enforcer and at the end, do a donut around the rocks in the big sandy area on the right side of a big turn.

To Drift, tap left and let go of the Right Trigger briefly and tap the Left Trigger while turning at a decent speed. You can then hold the Right Trigger when you start drifting and the direction you wish on the Left Stick to do a donut and keep drifting around one of the big rocks.

You do not however, need all 2000 drift points in one drift. It is fairly easy to get 2000 points before the end of the event without doing donuts or you can just turn around and build up some more points before finishing the event.

This can be done online.

The easiest way to do this is in multiplayer. Set up a friends only race and make sure you use the Subaru. I believe the course is "Escape to the Beach". Just drive until you reach a shortcut on the left that takes you to the open beach area. Keep drifting until you have 2,000 yds. Finish the race and POP!!!

Similar to MissJaded solution above go to Mission Beach under SCPD and there is a shortcut that cuts through the sand/beach area and you dont have to worry about hitting anything and can drift 2,000 points in one drift out there. Like previously stated don't worry about the racers finishing I was able to catch up or just take out one if you want before you get to the beach area.