A necromancer deals with the dead rather than the death rune.
So, he or she might have spells to raise the dead, create undead, speak to the dead, cause death, cause fear and so on.
Someone associated with the death rune might be a warrior or a butcher, or something undead, not necessarily a necromancer.
It might be interesting to ask your player how he wants to develop the necromancer, what he sees a necromancer as being, what cults, if any, he worships, how others see him and so on.