How would you track a player kill?

You would have somewhere storing each players deaths and kills. I usually use a table in a server-side script somewhere. Something like:

local Stats = {}

Then when a player joins simply add another table that is the players name such as:

game.Players.PlayerAdded:Connect(function(Player)
    Stats[Player.Name] = {}
    Stats[Player.Name]['Kills'] = 0
    Stats[Player.Name]['Deaths'] = 0
end)

Same for when they leave:

game.Players.PlayerRemoving:Connect(function(Player)
    --Any save code here
    Stats[Player.Name] = nil
end)

Tracking player kills would go something like this. If you were going to have the weapon be a gun, each bullet should have a value of the person who shot it. Then if that player dies, grab the value and increment that players kills by 1. So a little bit of pseudo code would be like this:

local Event = someEventFiredWhenBulletSent

Event.OnServerEvent:Connect(function(Player, Bullet)
    --Draw bullet here, get humanoid, take damage, etc, see the attached link at bottom of answer
    if humanoid.Health <= 0 then
        --Give person shot a point towards death
        local Victim = game.Players:GetPlayerFromCharacter(humanoid.Parent).Name
        Stats[Victim]['Deaths'] = Stats[Victim]['Deaths'] + 1
        --Now give the shooter a point for killing
        Stats[Player.Name]['Kills'] = Stats[Player.Name]['Kills'] + 1
    end
end)

See: http://wiki.roblox.com/index.php?title=Making_a_ray-casting_laser_gun as a good example for drawing rays, getting the person who was shot humanoid.