How do I make an ImageLabel show the player's current in-game character?

I managed to figure out how to set someone's avatar as the image in an ImageLabel, but how do I do the same thing, but with their current in-game character rather than their avatar? Because my game will eventually have a character customization system, and I want it to show the character that they actually play with.

Here's what I've got so far:

-- Player Stuff
local player = game.Players.LocalPlayer
local char = player.Character

local charAppearance = 'http://www.roblox.com/Thumbs/Avatar.ashx?x=352&y=352&format=png&userid=%d'

-- Gui
local frame = script.Parent

while wait(1) do
    frame.Image = charAppearance:format(player.UserId)
end

By the way, this is a local script inside of the ImageLabel I want to change.

Any help is appreciated!