the achievement isn't too difficult to get, only the wolf ring and the darkmoon is annoying.
The rest I gathered pretty fast once I got to NG3. Picking up any ring you come across beforehand helps of course.
If you wanna get every ring then do what i did, use ALT + F to use find on this page and type in each area like "Farron" or each bonfire. Don't stress and gather what this page says and unlock each area normally and gather the all the rings before going to the next.
Once you have gathered everything you'll probably come across what happen me and still don't get the achievement. that is because I missed 3 rings but have all the rings in your inventory (and no dups for easier handle) and have it sorted by default since this page is so as well. Go after each section and check off "okay I got life ring, life ring +1, +2, but not +3, ok I gotta find +3."
and that's it. I also recommend doing this with the other achievements as well (with the checking off, not going to NG3). But I recommend farming the ghrus before going to the next circle since the drop is the same but they get tougher/annoyingly tougher. Farming for the darkmoon covenant I also recommend doing before going to NG2 and not to level up to high so you might still get summoned (Good luck, worst RNG)
(Or you can go meta level, I personally got summoned more when I was meta lvl but I could've gotten lucky, got summoned four times in some hours which is worst then farming in my experience, goes for both farron covenant(didn't even get summoned once) and darkmoon covenant.).