If a person is dead, who is responsible for their credit card debt incurred during the time they were alive? What if he was pronounced dead at 3:00 PM, and the purchase on their card happens (or get posted) 3:01 PM?
Unsecured debt, like credit card debt, will be paid by the estate. If not enough money exists in the estate (including selling property/etc.) to satisfy the debts, then the debt will be cancelled - it is not continued on to the heirs/etc.
However, if you have a joint card (with a spouse, parent, child, etc.), both holders of that card may be separately liable - meaning if one dies, the other will be liable for the debt if the first's estate doesn't pay it off.
As far as if a charge posts after the death - likely depends on if it was fraudulently used. If the person owning the card made the purchase and timing just was off, then it'll be just as if they paid beforehand. But, if the creditor can prove that someone else authorized the charge other than the decedent/cardholder, then that is a fraudulent act and could be met with criminal penalties as well as a civil lawsuit.
See this Bankrate article on inheritance and credit card debt for more information.