Korea requires an FBI background check, and their processing times are between several months if you go directly through the FBI. We recommend instead going through Accurate Biometrics. They accept fingerprints through the mail and process requests in 24 hours. This is their website and list of instructions to submit fingerprints through the mail https://accuratebiometrics.com/nationwide/services/by-mail. In order to submit your fingerprints you need to print out two copies of the FD-258 fingerprint form on heavier paper/card stock and go to your local police station to have your fingerprints taken. Follow the instructions listed on Accurate Biometrics' website.
On the FD-258 fingerprint form the only required fields are name, date of birth, sex, race, signature of person fingerprinted, signature of person taking the fingerprints, and the date. You can see a form with the highlighted sections if you go to https://accuratebiometrics.com/nationwide/services/by-mail and go to section two and click on "Print two standard fingerprint forms (FD258) from the following link or obtain FD258 fingerprint cards." Make sure you have two FD-258 forms filled out and decorated with your fingerprints because sometimes your fingerprints just aren't pretty and they'll pick the best set to use.
Under section one "Complete the Application Form" and "Click here to download the form," you'll get the application form that will accompany your fingerprints. Under "Applicant Information" near the top there is a section under "phone number" for your address. It should be the address where you're currently residing and it can be an international address. You need to select the option to receive a paper copy in the mail with whichever shipping speed you're comfortable with.
If you're living overseas, it's important to do the following step. They won't mail your fingerprints to anyone but yourself (or an attorney with a bunch of extra paperwork). So do NOT put "care of" or anything like that. The section with the mailing address is at the bottom of the form. Send it addressed to yourself to an address within the USA of someone you trust who can accept your mail for you but make sure it's addressed to your name and not your trusted person.
If you go to https://accuratebiometrics.com/nationwide/services/by-mail it's all there for you. They are also really nice on the phone and happy to answer questions.
To get the documents apostilled, we recommend using US Legalization or US Authentication which can handle the process for you.