Kel'Thuzad is the final boss in the Wrath of the Lich King raid Naxxramas. Like all the other bosses in the raid he is fully soloable, but do to how some of his mechanics work, it's not as straight forward as the other bosses.
Many players have reported running into a bug when fighting him, where the encounter suddenly resets after a little while. This is caused by the fact that there are no mobs in the encounter to aggro you, so it's easy to circumvent.
In order to solo Kel'Thuzad, never leave the big, green circle in the middle of the room during the first phase. This is when waves of zombies and other necrotic enemies are coming towards you. Just let them come to you, and kill them when they are getting close.
Don't run to the small rooms in the edges of the encounter, and absolutely don't AoE all the enemies in those room. This is what causes the bug that resets the encounter when there are no more enemies to aggro you!
After a few minutes of killing enemies that come towards you, Kel'Thuzad himself will come to attack you. At this point you just nuke him down, and you will win the encounter.
So it's not very difficult to solo him as long as you just stay put in the green circle in the middle of the room.