Certainly not the fastest way, but if you have a solid internet connection and some patience this will work for you.
This method takes about 10 hours of racetime in 1 race. The good news is that you can walk away for hours at a time. When I did this my total race time was over 24 hours.
Get 7 friends to join a this race:
Gametype - Circuit Race
Track - Nordschleife
Number of Laps - 75
Damage Type - Cosmetic
Tires and Fuel - Off ***VERY important!!
I used a 2004 Audi #5 Audi Sport Japan Team Goh R8. Any R1 car will do. (Edit: Make sure its a 10.0 Rarity car.)
Suggested Line - Full
Anti-Lock Braking - ABS Off
Traction - TCS On (Makes getting back on the road easier)
Stability - STM Off
Shifting - Automatic
With 8 players it will say you get a baseline of $825,000. Wait 10 seconds or more and your friends can all drop out of the match leaving you alone.
This is the Nurburgring Circuit. Its almost 13 miles per lap. When I started I was completing laps (not including penalty time) about 8 minutes even. After 15 laps I was down to 7:45. In the end I was doing about 7:20-7:30 a lap. 75 laps will take about 10 hours at that pace. Comments are more than welcome if there are much faster cars to use.
At the end of the race you will receive $1,072,500. Achievement in 1 online race.
If you need to do a silly thing like sleep, eat, or go to work/school and you have a solid internet connection, and a Plug'n'charge cable, plug it in and put the controller down. As long as your Xbox Live Connection is uninterrupted your game will be uninterrupted when you get back.