You can probably do it online in your "my account" area, if you have one. Otherwise, write a simple letter stating you wish to change your billing address. Current address: 100 State Street, Chicago, Ill, (or whatever your current address is) and the NEW BILLING ADDRESS: 400 Jones Avenue... Etc. Sign your name and include your account number. That's all you need! Or course credit card companies are notorious for screwing things up, so it may take months for this change to occur.
Just simply go to the bank and ask them that there will be a slight change of your account regarding your new address ...that's all..
Credit Cards sample letters
Credit Cards sample letters