While black onyx does exist naturally, it's very rare. As a result, most commercially available "black onyx" is actually agate that has been stained black. Whether you've got a genuine black onyx ring or a stained agate ring that was labeled as black onyx, the cleaning process is the same. You should avoid using any cleaning agents on the ring, even a mild soap, because they may accumulate in the porous stone and discolor it over time.
Dampen a cotton cloth in water and wring it out so that it's moist, not wet.
Wipe the onyx portions of the ring clean with the damp cloth.
Dry the onyx portions of the ring carefully with a clean, dry cotton cloth.
Polish any metal settings or elements in the ring with a jewelry polishing cloth, taking care not to polish the onyx portions of the ring.