Just go to your local (main) police station. I got one in 2007. It was free (then) and took about 10 minutes. All that was required was my ARC and they took a thumb print.
It's still free, takes about the same time and all you need is your ARC. I've never had the thumbprint taken and I think they may have stopped doing that a few years ago.
The Korean for the document you want is 범죄경력회보서. I've never had this done myself, but I believe you can have it issued in English.
It's worth re-emphasising that you need to go to a main
police station. If you live in a metropolitan city, it'll be the police station for your -gu. In smaller cities, it'll be the city police station. But it is all incredibly easy, especially compared to the insane amounts of time and money required to get the equivalent piece of paper in many of our home countries.