How quickly you can double your bankroll depends on various factors. As a relatively simple example, let's say you play SNGs and have a 25% ROI. That would mean you'd be up $100 after playing tournaments you paid a total of $400 to enter. The number of tournaments would depend on the buyin level(s), and the length of time would depend on how much / often you play.
That said, you win money by playing where you're +EV. I'd start low and see how comfortable you are, and move up according to that as well as your bankroll, not just the latter, which seem to be talked about a whole lot more.