Assuming you are the executor of a will that is being administered in the United States, the law provides a fairly forthright procedure for notifying creditors and potential creditors.
First, you must send notices to all creditors you know about, as gleaned through a search of their personal papers and bills.
And second, you must publish a notice--called a Legal Notice--in the local newspaper where the person lived and was likely to incur debts. The purpose of this is to notify potential creditors--those you don't know about directly--of the death and their need to alert you about debts the deceased person owes them.
An anonymous caregiver
You will need to send certified copies of the death certificate. You will probably need many of them. I think there is some kind of opportunity to get many copies for cheap early on. To get them later on, can cost $5 or more each.
Posting a legal notice in the newspaper is a necessary formality. Most creditors are corporations that don't read local newspapers, but most companies won't cooperate with you unless you have made the legal notice, even if you have sent them a legally certified copy of the death certificate.