What part of Seoul are you in? Theres a good few of us scattered throughout Seoul, and chances are somebody would be able to recommend a gym close to you.
In terms of outdoors, theres lots of artificial walls around Seoul that are outdoors, that will be hit up randomly throughout the summer, so keep on eye on this board.
Once it warms up a little more, (maybe in 2 weeks), there will be groups going to real rock around the country. You'd be more than welcome to join. In terms of Seoul, there is Ganhyeon, which is 2 hours east by train and Surisan, in the very south of Seoul, which are probably 2 of the most accesible.
Best bet is to come along on one of the get togethers early on in the season, and go from there. Everybody is really chilled, and usually only happy to give back to the climbing community by helping each other out.