How do I make a billboard stay above the head when they respawn?

You can use the CharacterAdded event to achieve this effect in a script in Workspace.

game.Players.PlayerAdded:connect(function(player) --Defining the player
	player.CharacterAdded:connect(function(character) --Defining the character, triggers when the character respawns
		billboardgui.Parent = character.Head --Make sure to define what billboardgui is first!
	end)
end)

Of course, this is just an example script, feel free to change the code to meet your needs.