I had to do this last year! It's not hard if you're in the country, but outside I'm not sure.
There is more than 1 type of police check. The kind you need (the kind that can be apostilled) takes about a week - they might be able to do 3-4 days if you ask nicely and explain the situation. It has to be requested at the central police station. In Busan, this is the one at the City Hall subway station. I'm pretty sure the report was for free. If you need more than 1 copy, you have to fill out the application that number of times.
I'm not sure if you can order the police check by proxy or not, but it's possible that a lawyer's office could help you.
The apostille is more tricky - it can only be done in 1 place in Korea, in Seoul at the "MoFA" (Ministry of Foreign Affairs). Luckily, since the police check is a government-issued document, you can get the apostille put directly onto it. It's really cheap, around W1,000 or W1,500 per apostilled document if I remember correctly (that's per document, not per page).
The apostille can be done by proxy, but you have to send the original police check document to that person and they visit the MoFA in person. Turn around time for apostille is a couple hours IF you visit in the MORNING (they'll tell you to come back around 13:00)! If you go in the afternoon, next day service.
The other good news is that the type of police check you'll receive is in Korean AND English - better for using abroad!
Any other questions, feel free to use the email contact form and ask me directly. We ran around quite a bit but got it all done, and were able to use our documents in a couple other countries successfully :)