To begin with the process, you as a credit card holder must first ensure that all outstanding dues on the credit card are fully paid. As without it, banks will not accept and honour your application for credit card cancellation. Thereafter, you can get in touch with the call centre of the bank and place the request and make a record of all such details including service request number, date when the request for cancellation is placed with the concerned authority. For future follow up, you can also note the name of the representative with whom you have placed the request. Request for credit card cancellation can also be placed by visiting the bank branch.
Also, a letter in writing has to be sent to the credit card issuing bank, mentioning that you have placed the request for credit card cancellation. And after receiving their confirmation on the matter, you can cut and discard the credit card. Some banks may even ask you to provide the credit card pieces.
However, in case banks see any signature mismatch or overdues left uncleared, banks may cancel the request for credit card cancellation. In this case, you need to follow up with the bank, until you receive the final acknowledgement for credit card cancellation.