This ring achievement (and all the others) is most easily achieved in the Nightmare zone unlocked at the very start of the game. This zone can give you 900-1000 rings per trip through. Alternating between this level and zone 1-1 generally allows you to get rings as quickly as possible.
This is a useful strategy not only for the collect X rings achievements, but also the "Complete the level with X many rings" achievements as well.