Billboard GUI Issue?

camera = game.Workspace.CurrentCamera -- camera
part = -- location of the part containing the BillboardGui
game:GetService("RunService").RenderStepped:connect(function() -- refresh every 0.0167 seconds
    if (camera.CoordinateFrame.p - part.Position).magnitude < 20 then -- checks if the distance is smaller than 20 studs
        part.BillboardGui.Enabled = true -- makes it visible
    else
        part.BillboardGui.Enabled = false -- makes it invisible
    end
end)

This script will check if the camera position is in the limit of 20 studs. If you want to check the character, then just change this

camera = game.Workspace.CurrentCamera -- camera
    if (camera.CoordinateFrame.p - part.Position).magnitude < 20 then

for this

character = game.Players.LocalPlayer.Character
    if (character.Head.Position - part.Position).magnitude < 20 then

. Magnitude is very useful.