Pretty simple. You can complete matches in the Mystery portal to earn Rainbow stars. You earn three Rainbow stars for a win and 1 Rainbow star for every completion. Hope this helps, currently working on this myself. Happy Hunting.
Note: selling your normal stars to the seagull in the Town Hall for Rainbow stars does not count towards the achievement.
Edit: There must have been an update since the day the DLC dropped. Wins now grant you 5 Rainbow stars and completions grant you 3 Rainbow stars.