In order to get promoted, your crew needs to be in the green area of the 'tier status' bar. There are 2 things that contribute to this, your crew's score percentile and xp percentile.
Percentile is where you're ranked for the particular challenges. If you're in the top say 2% for score, you're in the 98th percentile. It takes both of these percentiles and averages them. As long as you're in the green area when the challenge ends, you'll rank up and get the achievement.
The easiest promotion would be starting in a crew at bronze and going to silver.
Edit: People are reporting issues with this and other rivals achievements unlocking.
Edit2: If you go to the Rivals Page and hit Orange/RB, it should pop the achievement. Note this only seems to work during the 'cooldown' period between challenges.