Join a game of The Eliminator and just ignore all the other drivers and only concentrate on finding Car Drops, which are areas where a drone is flying with purple smoke. Press the button to claim the car and you'll get whatever car the drop is. Keep driving around and claim 4 more Car Drops in the same session and you'll get the achievement. They tend to be near buildings and landmarks so avoid the wide open areas.
One of the best general areas for car drops is the north side of the map. I would drop at Mortimer Gardens (top left). Even if there are no drops there, moving north-east from there takes you to a very hilly and open area, making it super easy to pick out drops from the distance. I managed to get 38 cars in 7 matches this way!
Unless you're confident in your abilities or have a really fast car, I'd try my best to avoid letting other drivers challenge you because if you lose the head to head you're out. So just give everyone a wide berth and keep searching for those smokey drones.
The upgrade cars you get from winning head to head races do not count. It has to be one of the drones.