For her to go without being command sponsored: 1. She would need a visa through the Korean government authorizing her to stay there for a year. This would require her to either get an Overseas Study (D-2) visa, or a Foreign Language Teaching (E-2) visa.
"The Foreign Language teaching (E-2) visa is issued to foreign language teachers who work in South Korea. Applicants are required to be native residents of a country whose mother tongue is the same as the language they will teach and they are also required to hold a bachelor's degree from that country. Applications are required to submit criminal background checks, health checks, sealed transcripts, verified copies of their degree, criminal record check, health check, consulate/embassy interview for first-time applicants, contracts and a fee to obtain the visa.
Finally, since she is not command sponsored, you would still be a geo-batchlor in the eyes of the Army. That means that you would be required to live in the barracks and would not get to live in the same house as her.
I'm not even going to get into the amount of money this all would cost you since the Army wouldn't be covering any of this. Just know this, just the plane ticket to fly your wife to Korea is going to cost 1,000 dollars by itself. As far as household goods go, good luck. I have seen some families pay thousands of dollars by simply going over the weight limit the Army covers. I can't imagine how much it costs the Army to ship thousands of pounds of your stuff to Korea.
If you want a final laugh, this lady made a website all about how she was going to go to Korea without command sponsorship. If you read the comments at the bottom she learned that A: No, it's actually not easy. and B: Being American is not a reason for any country in the world to just let you live there. She ended up getting command sponsored. The special stamp that she is talking about that allows you to "extend" your tourist visa, is a SOFA stamp only given to command sponsored spouses. It's not just something you can get.