I have looked all over Google and everyone says he's in the Icecrown Citadel but I can't find him anywhere. Where exactly can I find him?
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.
The Lich King is the final boss in the raid instance of Icecrown Citadel. Icecrown Citadel is located within the Icecrown zone - are you sure you're actually inside the dungeon, and not simply in the zone with the similar name?
The Lich King is inside of the IceCrown Citadel, in the Icecrown Zone of Northrend. Once you clear the other quarters, the final bosses of each Quarter make it possible to kill the Lich King. You will find him in the Upper Spire. Once you get there you can use teleporter at entrance if you need to go get more people to help you with him. Once you walk in the upper spire and go through another portal type device, you will be in a room with Highlord Tirion Fordring, and the Lich King on his frozen throne. Speaking with Tirion will start the The Lich King Fight