The BMX Challenge will become available at the beginning of the game, after completing the Big Smoke mission. You can start the challenge at the BMX park in Glen Park, Los Santos, but your Cycling skill will need to be at least 20% in order to start (100% is recommended for an easier time).
You'll have to collect all of the coronas before time runs out. You start with 10 seconds and an additional 10 seconds is added for each corona you collect. A higher Cycling Skill allows you to bunny hop higher by holding and releasing L1, which makes it a bit easier to get some of the higher coronas. Start out by collecting all of the easy coronas along the ground, then head onto the harder ones like the ones on the halfpipes when you have accumulated some time. If you fall off the bike, you have 25 seconds to return to it before you fail.