Each time you log into the game for the first time that day you will receive credits from your Drivatar. Your Drivatar participates in races in other people's games and this way it earns credits.
In the message center you can see how many credits your Drivatar has earned the previous day. On average this will be somewhere between 3.000-8.000 credits, although on occasion this might be higher or lower. For me it averaged at about 4.000-5.000. Based on these numbers, the achievement will take between 8-15 days to get.
The most important thing to getting this achievement is logging into the game on a regular basis. Unlike previous Forza-games that used this system, it seems that credits do accumulate over time. If you don't log in for several days and then go back to the game you will receive more credits than when you log in on consecutive days. There doesn't seem to be much difference between logging in daily and logging in every couple of days in terms of credits you receive.
Although not confirmed by Turn10, the more friends you have that play Forza Motorsport 6, the higher the daily payouts will be. Your Drivatar mostly appears in your friends' races, so the more they race, the more credits you get. This can generate quite a lot of credits (up to 10-20K a day). If you want the achievement fast, make sure you have a lot of friends that play the game. Otherwise it will accumulate to the necessary amount over time. (Thanks to Sol76 for more info regarding payout in relation to amount of friends)