It takes a combination of:
- a minimum number of ranked rounds played at your current tier, and...
- a minimum scoring average
For example, to move from Tour Pro to Master, you need at least 30 ranked rounds played as a TP, and an average below 67.
To move from Master to Tour Master, you need 40 ranked rounds and an average below 63.
To move from Tour Master to Legend, you need at least 50 ranked rounds and an average below 61.
Your average resets to zero every time you change tiers.
Your average is also not really your average, but the average of the best x number of games at your current tier. (Best 30 games for TP, best 40 games for Master, best 50 games for Legend, etc)