Instructions
Start your Xbox 360 and log into your Gamertag. By default, your system should be on the Xbox Dashboard. If it is not, navigate there.
Click the "Systems" tab from the Dashboard. A menu will appear.
Click the "Language" menu and press the "A" button on Korean. The Korean option is the second-to-last option listed. This will change the language of your console and all the text that is displayed to Korean.
Press the "A" button again to confirm the change to Korean.
Press the "Xbox Media Guide" button in the center of your control and press the "A" button on your Gamertag, which will still be in your original language.
Press the "A" button on the second option listed, which is the Edit Profile menu. From here you can change your motto (fourth option listed), name (fifth option listed), location (sixth option listed) and your biography (seventh option listed).
Press the "A" button on the option you would like to change. A text window will appear, except the characters and keyboard will be in Korean.
Press the right trigger to view additional character sets, including many of the Xbox symbols.
Press the "Start" button to save your changes. When you're finished, just convert your language back to English, just as you did in Steps 2 and 3.