Why is my local script working the first time but after they die it stops?

Your repeat until loop is waiting for the character, and when the character ~= nil it executes the code below. The problem is though that it will only repeat once though.

Fix:

repeat wait() until game.Players.LocalPlayer.Character
print("safdsjhfl;ksajd;lfkjasd")
while true do

if game.Players.LocalPlayer.Character.Beep.Value == 0 then
        game.Players.LocalPlayer.PlayerGui.ScreenGui.Dave.Grenade.Text = "x 2"
        end
game.Players.LocalPlayer.Character.Beep.Changed:Connect(function()
    print("why?")
    repeat wait() until game.Players.LocalPlayer.Character
        if game.Players.LocalPlayer.Character.Beep.Value == 1 then
    game.Players.LocalPlayer.PlayerGui.ScreenGui.Dave.Grenade.Text = "x 1"
     else if game.Players.LocalPlayer.Character.Beep.Value == 2 then
        game.Players.LocalPlayer.PlayerGui.ScreenGui.Dave.Grenade.Text = "x 0"
end
    end

end
end)