Mutliplayer scoring system

When a gameobject is destroyed, i want the players score who destroyed the gameobject to increase only, currently if anyone gets a kill, it updates everyones score.

How can i determine which player got the kill? so it only adds to the score of that player for that kill

How is the object being destroyed? What code is being called, and by whom, to destroy it?

Without knowing more of how your code is structure, you should know which object triggers the destruction of the target, which should let you determine the player responsible.