You can mark a pet as deceased by opening the pet card and clicking the Deceased box on the right panel.
You can opt not to show deceased pets on the client explorer in File, System Options, Global tab.
If you do want deceased pets to display on the client explorer, the light bulb icon will be dim. So in this example, Yukon is a deceased pet.
You can get a list of deceased pets in Reports, Lists, Deceased pets.