Why does the Spree leaderstat not get reset?

You are only getting the first character. Once the character dies and another character is created, it doesn't care about the new one. Connect to the character with player.CharacterAdded

local PPS=game:GetService("PointsService")
game.Players.PlayerAdded:connect(function(player)
	local counter=Instance.new("NumberValue",player)
	counter.Name="Counter"
	counter.Changed:connect(function(item)
		if counter.Value>=10 then
			counter.Value=0
			player.leaderstats.FPoints.Value=player.leaderstats.FPoints.Value+1
			PPS:AwardPoints(player.UserId,1)
		end
	end)
	player.CharacterAdded:connect(function(character)
		character:WaitForChild("Humanoid").Died:connect(function()
			counter.Value=0
			player.leaderstats.Spree.Value=0
		end)
	end)
end)

It also seems that you're using leaderstats and its elements but not creating them in this PlayerAdded connection. I recommend only using one connection to PlayerAdded.