How would I make a character gain exp when another character kills them?

The best way to create a function that does something upon a death is by using the .Died event of a Humanoid. How this event works is that when a Humanoid dies, the function is carried out. To create a function that would give a player EXP once he killed another player, you'd do something like this:

script.Parent.Touched:connect(function(touched) --In this case the parent of the script is a tool
    if touched.Parent.Humanoid then  --Script checks to see if it is a human
        touched.Parent.Humanoid:TakeDamage(10)
        if touched.Parent.Humanoid.Health == 0 then --If the opponent who was hit was killed then this will run
            script.Parent.Parent.EXP = script.Parent.Parent.EXP + 1 --Adds EXP
        end
    end
end)

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