There is a Donor Status field where you can mark a donor as "Deceased". This field however is not ideal for a Household profile, because you will not know which member of the household is deceased.
In that case, you would remove the deceased donors name from all of the fields within that donor profile, then make a Donor Note stating that John Doe passed away, etc.
To change the Donor Status to deceased, edit the donor profile and scroll down to the Development Details section. Here you will see Status field. Choose ‘Deceased Donor’ from the dropdown list and then click Save.
Changing a Donor's Status to Deceased will NOT automatically exclude them from lists, reports and mailings.
However, you can use the Deceased Donor Status as criteria in the "Don't Show" tab when creating a SmartList to exclude any Deceased Donors from that list.
Also, when doing a mailing, if there are Deceased Donors on the list you are using for the mailing, you will see a warning in red at the bottom of the Recipients screen notifying you that some recipients on this mailing are deceased. You can then click on the Uncheck option and choose to 'Uncheck all deceased recipients'.