Depending on your Club's performance in Club Battles, you can score between +75 and -75 per battle. This score is then tallied up with all your other Club Battle results in that Season to create your team's total score.
There are three types of Club Leagues: OPEN, AMATEUR, and PRO. After a season ends, active clubs are divided into Open, Amateur, and Pro Leagues based on their performance during that season.
Clubs who performed exceptionally well in that season may move up from OPEN to AMATEUR, where they'll be pitted against tougher Clubs. The same is true for moving from the AMATEUR to PRO League. On the flip side, if a Club lost several Battles during the Season, they run the risk of moving downwards in League and Leaderboard.
You can see this score on the different League Leaderboards beside each club. Spots are limited, so you'll be in competition with other Clubs for the League spots.