How do I get this avatar customizer to work?

You are putting the piece inside the player. Try it inside the player's character.

LocalScript

local RS = game:GetService("ReplicatedStorage")
local Player = game.Players.LocalPlayer

script.Parent.MouseButton1Click:Connect(function()
	local Halo = RS:WaitForChild("Halos"):WaitForChild("Halo"):Clone()
	local Character = Player.Character

	Halo.Parent = Character
	Halo.Position = Character.Head.Position + Vector3.new(0,5,0)
	local Character = Player.Character
end)

However, the piece will be visible only to the player, not to the other players on the server, because it is a function in a LocalScript. Use a remoteEvent in RS, and a Script in SSS, so that all players can see the piece.

LocalScript

local RS = game:GetService("ReplicatedStorage")
local RemoteEvent = RS:WaitForChild("RemoteEvent")

script.Parent.MouseButton1Click:Connect(function()
	RemoteEvent:FireServer()
end)

Script in ServerScriptService

local RS = game:GetService("ReplicatedStorage")
local RemoteEvent = RS:WaitForChild("RemoteEvent")

RemoteEvent.OnServerEvent:Connect(function(Player)
	local Halo = RS:WaitForChild("Halos"):WaitForChild("Halo"):Clone()
	local Character = Player.Character

	Halo.Parent = Character
	Halo.Position = Character.Head.Position + Vector3.new(0,5,0)
end)

I hope I helped, if it goes wrong, could you tell me what broke the Script? That way I can help fix it. :)