The first thing you should do when someone dies is to preserve the information that will be needed to conclude the succession. This information consists of information about the decedent, such as the date he died, where he lived when he died, whether he executed a valid testament before he died, who are his heirs or legatees and what property and liabilities did he leave behind.
A good practice is to gather together in one place all of the bills and bank statements that are received in the mail over the next 30 days or longer. If the decedent had any stock, bonds or other securities, it would be useful to retain a listing, such as a newspaper listing, of the quotes for the day he died.
It is also important to retain all of the bills for the funeral and burial. These bills are considered "administrative expenses" of the succession, and are treated differently from all other bills which the decedent had at the time he died. If the decedent had pre-paid his funeral expenses, you should retain those records.
If the decedent died testate, it is important to secure the original testament. If you try to probate a copy of a testament, the judge will assume that the decedent destroyed the original with the intent that it be revoked.