Here's kind of what I'm thinking.

Rank 1
-any player with 3+ wins this week who won more battles than they lost is promoted to Rank 2

Rank 2
-of the players with 5+ wins this week, the 4 players with the best winning percentages are promoted to Rank 3
-any player with 0 wins this week is demoted to Rank 1
-of the remaining players, the 4 players with the worst winning percentages are demoted to Rank 1

Rank 3
-of the players with 7+ wins this week, the 2 players with the best winning percentages are promoted to Rank 4
-any player with fewer than 2 wins this week is demoted to Rank 2
-of the remaining players, the 2 players with the worst winning percentages are demoted to Rank 2

Rank 4
-of the players with 10+ wins this week, the one with the best winning percentage is promoted to Rank 5
-any player with fewer than 3 wins this week is demoted to Rank 3
-of the remaining players, the one with the worst winning percentage is demoted to Rank 3

Rank 5
-any player with fewer than 5 wins this week is demoted to Rank 4 (obviously this doesn't count until there's more than 1 person in Rank 5)