SOME IMPORTANT NOTES:1) The objective is just to touch on any cop cars active during chase or in roadblocks to count, as trade paint. Hitting and disabling cop cars will count towards your progress, obviously.
2) You must ram at least a different cop car once. Ramming the same cop car multiple times will not count.
3) The HUD on top-right of screen only shows dispatched and disabled cars. You must to consider at least 15% of loss during the chase. It means that some cop cars will be disabled during the chase just because they crashed by accident, without any contact with your car. Read note 4 bellow to understant it better.
4) I got the achievement after an 1h35min chase. I've decided to end the chase with 139 cop cars dispatched and 133 disabled. It means that I propably rammed just 110 cop cars or less.
5) Trying to get this achievement after completing the story is much harder. Start a fresh career file, obviously on the same profile, if, this profile is the main profile of your entire game progress. When you get this achievement, you get the Crusher reward card from career as well. If you plan to get the Reward Card Hustler achievement, you need to do everything on your main profile of the game.
STARTING A FRESH CAREER ON THE SAME PROFILE:You must to play all intro part, including the training process. After the training, you'll have just three race options. Just play until the nitro and tires upgrades become available for you. Purchase the nitro and tires upgrades only. Set the balance of your nitro focusing in speed and your tires for max grip. This way your car will become heat level 1. During the chase your car will only reach heat level 2. On heat level 2, only regular cop cars will chase you, it means no spikes strips on roadblocks and no rhinos (SUVs) chasing you.
TURNING OFF AUTOSAVE:You must to turn off autosave feature to avoid losing your car in case you fail during the chase by the cops.
To turn off autosave go to: MAIN MENU > OPTIONS > GAMEPLAY
FREE ROAM AND ACHIEVEMENT PROGRESS:With everything set, go to FREE ROAM and start hitting some vehicles or environment items to get cop cars to chase you. Remember to look for the flashing red arrow on the map on bellow-left of the screen. This red arrow is the cop car, go there to start the chase.
I found this place in Kempton (south of map) on the picture bellow, the best place to control the chase with less risk of something going wrong. Keep runing in counter-clockwise on this red area marked on the map. I've done this way twice with success.
About 3 to 5 cars will be dispatched to catch you before the game calls for backup. Try to disable as many cars as you can until the next backup. Keep yourself away of "dynamic breakers" and "landmark breakers" to avoid cooldown status.
Backup will be dispatched when you have 1 or 2 cop cars chasing you. Keep running until the next backup arrives. Keep a safe distance of cop cars, to not be captured and short enough to not enter in cooldown status. If for some reason the cop car stop to chase you, try to go back near of them to avoid cooldown status and when they start to chase you again, escape keeping a safe distance.
Do not avoid roadblocks, try to hit all cop cars at roadblocks always you see them. This way you can reach the achievement goal faster. If possible, always use speedbreaker combined with nitro on roadblocks. When you use the speadbreaker, the game turns into slow motion, making easier the job of hit the right place of cars on roadblocks. Never hit cop cars at roadblocks from behind or front. Always try to hit the cars laterally between bumper and fender. Depending of the positions of cop cars at roadblocks, you can hit two of them at the same time.
Good luck!