This is grinding leveling achievement of the game. You must reach level 35 online, which happens at 250,000 fans.
First, do the Honorable Driver achievement and get that white hexagon next to your name. This will net you 20% extra fans per race from now on, and hopefully make your journey a few hours shorter. If you play games without "Impact Rating" and race cleanly, you will keep this rating.
I recommend playing Team Based modes in either Pro Tour or Jam Session. These give you the best chance at winning, and a good reward when you do. The trick is to find events that are only 1 race (round), as doing more than 1 round is just a waste of time for the same amount of rep. An average player will earn about 750 rep per race (event), up to 1250 rep per race. At those paces, you are looking at somewhere between 200-333 races total online for this achievement. At an average 4 minutes per race (with menu + loading), you are looking somewhere between 14 and 22+ hours.
This achievement can be boosted. You will need 8 people and it will take approximately 6-7 total hours. The first thing to do is make sure everyone has "honorable driver" status to maximize on the fans. It is ok if you don't, but it will make a difference of 1,600+ fans/hour. Next you should make 2 groups of 4 people, and have everyone in the same party chat. Ideally, make the best connections should lead the 2 groups. Both groups should now search for Pro Tour, Gymkhana, and should have no problem finding each other. Once in the same room, you start the match and make sure that nobody scores, but make sure everybody moves their car from the starting position. At the end, everyone is to have 0 points, resulting in a tie.
Post Game Results:
1st Place Finish: 1,000 fans
Repuation Boost: (+20%) 200 fans
Team Victory: 500 fans
Total: 1,700 fans
Each game takes 1:30, and loading times and screens in between matches over a minute and a half. At this rate you should get around 27,000 fans per hour, and 8-9+ hours of boosting total should net you the achievement.