The Lich King can actually be found in many different places, including the Death Knight starter zone, several 5-man dungeons and appears as a result of many quests in Icecrown Citadel. If you level normally throughout the “Wrath of the Lich King” expansion, you will encounter him on several occasions.

To actually battle him, you need up to 25 other players in the Icecrown Citadel raid instance. For him to properly spawn, all three wing Guardians (Sindragosa, Blood-Queen Lana’thel and Professor Putricide) must be downed. Doing so allows you to teleport to him through the center of the Citadel. I believe even if you somehow got up to the top of the Citadel, he will not spawn unless those preconditions are met.