You will need the Profile Deactivate permission to perform this action, at one of the person's campuses. It is campus-scoped, so a campus override can change it at one campus. See People for what the permission covers, and Campuses and organizations in Version 8 for how campus access works.
When a profile is marked as deceased, the person will be archived and will not appear on communication-related reports such as address labels, giving statement greetings, etc.
To mark a person as deceased, navigate to their profile and then go to Actions > Archive in the upper right.
Then in the pop-up, choose Deceased as the reason. You will also be shown a list of any involvement they had.
Tip: A profile archived with the reason Deceased stays archived when a contribution is credited to it. Every other archive reason unarchives the profile automatically in that case.