The Lich King is the supreme ruler of the Scourge, the faction encompassing the majority of undead on Azeroth.
Ner’zhul: The original Lich King, created by the Burning Legion from the spirit of the orc shaman Ner’zhul after the Second War (year 8) to eventually pave the way for their next attack on Azeroth.
Arthas Menethil: Perhaps the most infamous and well known, Arthas became Lich King when he merged with Ner’zhul atop the Icecrown Glacier shortly after the Third War (year 22). The Legion had been defeated by then and the Scourge was an independent entity. He reigned until the joint Horde and Alliance war campaign brought him down at Icecrown Citadel in Northrend.
Bolvar Fordragon: After the death of Arthas, Bolvar succeeded him to keep the Scourge under control. There must always be a Lich King!