Earning fans is done by playing shows in Go On Tour mode, which you can find in the Campaigns tab. By the time I finished my first run of the tour mode I had garnered about 175,000 fans, but you can easily earn more by doing the various bonus shows unlocked along the way. You can always just start another tour with a new band of course as this achievement is cumulative.
SIDE NOTE: Taking requests and doing encores doesn't actually increase your amount of fans, only the money earned from the gig. So don't feel the need to do either unless you're in need of some extra cash and/or want the Tour of Booty achievement.