It moves incrementally and also depends on whether you are playing on the mats or in Competition Mode.
When I say "incrementally" what I mean is that as you move up in belt levels, it takes more wins to get to the next color.
I have figured so far that it takes
3-5 White
4-6 Yellow
5-8 Orange
That is all I have done so far. Our sister site,
Club Penguin Fansite, might be a good place to learn more from others.