First, whose name is on the account? If your credit card is under your name only, your estate should take care of your debts. Your estate administrator will pay off all your debts, your credit card debts included, according to the order as required by the law. Remaining assets will be distributed to your heirs accordingly.
If your card is shared with another person; that is, a signatory to the card application, he could be made to answer for your debts. However, if the other person is listed only as an authorized user, he is not liable for the debts.
Second consideration would relate to your estate. If your estate is not enough to cover your debts, then that would be the end of it. Your heirs would not be liable for those debts. Your card company’s option is to write off your debt.
However, there are “community property” states in the US who regard assets and debts acquired throughout the duration of a marriage as joint property. In this case, the heirs would be liable to pay the debts if the estate would be insufficient to cover them. These states are Washington, Texas, New Mexico, Nevada, Louisiana, Idaho, California, Arizona and Alaska.