How to change the color of a character when they (re)spawn?

I'm working on a script that changes a player's colors to a random color every time they spawn. I have tried many variants, none of them having worked. Here is the latest variant.

--This is a LocalScript

player = game.Players.LocalPlayer

player.CharacterAdded:connect(function(character)
	player:WaitForDataReady()
	local color = BrickColor.Random()
	local head = character["Head"]
	local torso = character["Torso"]
	local rightarm = character["Right Arm"]
	local leftarm = character["Left Arm"]
	local rightleg = character["Right Leg"]
	local leftleg = character["Left Leg"]
	head.BrickColor = color
	torso.BrickColor = color
	rightarm.BrickColor = color
	leftarm.BrickColor = color
	rightleg.BrickColor = color
	leftleg.BrickColor = color
end)

Could somebody please point out what the problem with my script is?

There are three problems here:

WaitForDataReady has to do with Data Persistence and should not be used here, Body Colors overwrites the changes you want to make so you'll have to get rid of that first, and this likely won't run the first time because the character will (probably) load before the script is ran.

A better way to do this would be in a server script:

game.Players.PlayerAdded:connect(function(player)
	player.CharacterAdded:connect(function(character)
		character:WaitForChild("Body Colors"):Destroy()
		local color = BrickColor.Random()
		character["Head"].BrickColor = color
		character["Torso"].BrickColor = color
		character["Right Arm"].BrickColor = color
		character["Left Arm"].BrickColor = color
		character["Right Leg"].BrickColor = color
		character["Left Leg"].BrickColor = color
	end)
end)

Sometimes when the character is spawned, the model may be there, but the parts aren't. Use the CharacterAppearanceLoaded event.

game.Players.PlayerAdded:connect(function(plr)
	plr.CharacterAppearanceLoaded:connect(function(chr)
		local color = BrickColor.random()
		for _,v in next, chr:GetChildren() do
			if v:IsA("Part") then
				v.BrickColor = color
			end
		end
	end)
end)