By scrolling through your leagues' leaderboard you may see a PROMOTION ZONE on the top and a DEMOTION ZONE on the bottom. Make sure you make enough score during the league period, so that you are above the PROMOTION ZONE marker, to be promoted to a higher league. If on the contrary, you did a poor score during the league period, and you are below the DEMOTION ZONE marker, then you will be demoted to a lower league 

Keep in mind that the higher the League you are competing in, the more difficult will be the competition as more competitive players will be also competing to be Promoted to the next League Division.