The goal is to get into the 50,000+ bounty range without pulling the best laptime you can manage. You have to be fast, but you can't just run your best lap right away. It takes some trial-and-error to find that sweet spot.
Then, as ImaRobot said, you challenge the player immediately above you. Beat his time by a very small margin (if necessary, "sandbagging" your time by slowing down at the end), and repeat the process with the next player above you.
Example:
You set a laptime of 1'50.356. You challenge the player above you (1'50.355). You beat him with a laptime of 1'50.275. So you challenge the next player (1'50.274). You beat him with a laptime of 1'50.131...
Say the best lap you could possibly do with your skill level is a 1'45.xxx -- you can reap some big rewards inching your way down to that time. And even after you reach 1'45.xxx, there's nothing stopping you from improving on it; it simply becomes a true challenge at that point.