Who the Lich King is depends on when you are asking the question.
The original Lich King was created by Kil'jaeden from the soul of the orc Warlock leader, Ner'zhul, after he escaped from Draenor. Initially the Lich King, although given immense powers by the eredar demon lord, was trapped in Northrend in the ice crystal block called the Frozen Throne. This throne was both a prison and a method of focus. It kept the Lich King from escaping his dreadlord overseers, but did not limit his influence. After the Lich King corrupted Arthas and compelled him to shatter the ice of the throne, the Lich King's soul merged into Arthas' body. United they became the new Lich King and even more powerful than before due to the possession of Frostmourne.
The Lich King began with tremendous psychic powers with which he used to enslave undead created by the Plague of Undeath and turn them into his army of the Scourge. His power is such that he can psychically project his likeness and speak over vast distances. His reach in Northrend is so powerful, his form can be visible to adventurers he seeks to corrupt, but at farther distances such as other continents or Outland, he can merely speak in sinister whispers.