The journey rather than the procedure: what to settle before you book, whether anyone will speak English, paying, and getting home. What a treatment does is on its own page.
Follow-up is the part of treating abroad that nobody plans and everybody needs. Leave South Korea holding the clinic's Korean name and contact, an itemised receipt, a record of exactly what was used on you and by whom, your consent forms, and written aftercare. If a complication becomes a dispute, South Korea has a statutory agency that mediates outside the courts.
For self-pay skin treatment you generally need a passport, a consultation, and a way to pay — not residency, a Korean ID, or national health insurance. The larger questions are who will interpret, what is written down before you pay a deposit, and who you contact after you fly home. Settle those in writing rather than in conversation.
This site does not recommend clinics, so this is about method rather than names. Search where Koreans search rather than only in English, check the clinic against the register of institutions permitted to treat foreign patients, and know that a clinic's name carries a signal about its doctor's specialty. Any licensed doctor may legally perform these procedures — the question is whether you know which kind of doctor you have.