DiRT Showdown

If you don't already have it, this will also get you

Grab your 2nd controller (or borrow one if you don't have one)

Set up a splitscreen match as...

Competition - Individual
Discipline - Rampage
AI drivers - 0
Rounds - 1 round

Then go to Track Selection & choose...

Los Angeles
American Auto Bowl - no obstacles to negotiate }:o)
time of day/weather doesn't matter
10 mins (should only take 5, tops)

Car choice doesn't really matter, but I'd go with the Holbrook Growler for your main car - highest strength & power combo meaning you can do max damage & be less likely to wreck yourself by accident, & the Burton Beachcomber for the player two car - one of the lowest strength cars.

Drive player 2 car into the ring & destroy him with your main profile car, rinse & repeat four more times.

Achievement unlocked.

Whilst there is nothing wrong with the other guide here's how to do it properly in career with ease and is at least a little more enjoyable which does not require a second controller, though you will need a second controller for the split screen achievement.

Enter career mode lowest tier level, first rampage game type and pick a car with good strength and power, the map is an obstacle free bowl with 2 opposing barriered loops.

At the start wait on the ramp, you will be forced off shortly, after this hide behind the barriers in one of the two short loops avoiding trouble giving the drones time to do considerable damage to each other, once the timer gets down to a little over 2 minutes, say 2:20 its time to move, keep racing across the bowl one barriered loop to the other targeting any that have low health (small amount of remaining health bar red in colour) on the way. Because you now have a significant health advantage over all the drones a 5x streak is very obtainable, but not guaranteed first try. Use the boost button if you feel your going to get a good hit or to get out of trouble but try not to waste your boost as it does not build up as quick in rampage as it does in other race types.

PS Dont hate the guide for a legit method, if you want it easy first go, get a second controller and go with the split screen method and plus vote the other guide.