How do I delete a credit card
How do I get rid of a credit card on file and make it empty so I can update my games
On your computer's iTunes you should be able to edit your payment info by going into the Store > View Account menu option and logging into your account, and on your account's details page there should be a payment link. If you are doing it on your iOS device then tap on your id in Settings > iTunes & App Store (Settings > Store on iOS 5 and below) and tap on 'View Apple ID' on the popup and log into your account - that should also give you a payments link on your account's page.
Changing payment info : http://support.apple.com/kb/HT1918
If you don't get the 'none' option on the payment details screen : http://support.apple.com/kb/TS5366