After playing through the career on easy, and doing 6 time trial laps on the shortest track: Mythic, Utah, you'll be around 180 miles traveled having got the other 11 achievements.
You can check your progress be going into the leaderboards and press X to switch to local. Swith to the miscellaneous page using the bumpers and cycle through the 6 tracks with the left thumbstick adding up the miles on each page.
Now its a matter of travelling another 820 miles.
You'll want to use the fastest vehicle, the Trophy Truck, with maxed speed and acceleration.
This will take the least amount of time if you do time trials on the Chippewa Hill, Michigan track which has long straight-aways and no tight corners. Turn off road hazards. You can do around 120 miles per hour on this track with a maxed trophy truck so be prepared for 7 hours or so of extra driving.
If you prefer to leave the game running unattended while you're asleep or at work/school, you can rubber band the thumbsticks together and the right thumbstick down doing donuts at around 30-40 miles per hours. This will take a lot longer then driving yourself, on the order of 20+ hours. EDIT: You either need a wired controller and a plug and play kit on a wireless to do it this way.