Some are are only available as covenant rewards and can't be obtained otherwise. Most are found in static places throughout the world. In subsequent playthroughs, all of the rings from previous playthroughs are available as well, so you can't miss out on anything that you didn't get by going to the next playthrough.
The +2 versions (and single +3 version) can be found in playthrough 3 (again, in unique locations).
The +1 versions can be found in playthrough 2 (in unique locations). The basic versions of each ring can be found in the first playthrough. Any that appear more than once have multiple versions that can be found in subsequent playthroughs.
This image shows inventory order for all rings. Therefore you need all 70 basic rings AND all +1 and +2 versions (one ring has a +3 version, too). This achievement requires collecting all 107 rings, which includes all variations.
