To move to a higher league, you have to simply play well, win matches and rank high in your league. Then you might be promoted to a higher league. Same goes with demoting - play bad, get to the bottom of the ladder and you will be demoted to a lower league.
When playing matches, you might encounter players from almost any league i think. I personaly am ranked in a platinum league and played a few matches with gold or silver ranked players. You will see that sometimes the load screen reads "Slightly favored" or "Favored" above you or the oponent. That means one of you is considered better and the game thinks he/you should win.