Your current objective in the quest "Reign of the Black King" is to search for the Royal Crypts. As previously mentioned, it is important to left click on your follower's icon, and assign him the HEAL skill, a level 5 skill. In the battle against the Skeleton King, this skill we be useful.
You also get the following achievement for having the templar, Kormac, as your follower:
A Boon Companion
Acquire the Templar as a follower
The exit to the Cathedral Level 4 should be nearby. Take it, and descend through some spiral quest to fight some Harvesters (Knockback). There are also Skeletal Archers in this level, that attack you from range, and the usual Skeletons, Tomb Guardians, Fiends, and other undead. You will come across some Tomb Guardians performing a ritual, break their ritual. They will be joined by Harvesters and Skeletal Shieldbearers. If you die, you revive at your last checkpoint.
There are lots of crumbling walls in this level, so it is a good spot to get the achievement:
Bringing Down the House
Kill 10 monsters with the environment
There are also Chandeliers in this level, which you can cause to drop by cutting the rope. If you kill an enemy with one of those, you will get the achievement:
Light Entertainment
Kill a monster with a Chandelier
The fourth level of the Cathedral is a pretty big one, so you will have a lot to explore before finally reaching your exit. Open your map once in a while, to get some orientation. You should find a Scribe's Lectern, with a green story item: Lachdanan's Scroll Part 2 on it, which will be added to your journal. You might also encounter a harvester elite boss, such as Drekavac Filthpit. Break any wooden doors you see, because many of them lead to rooms with chests and treasure. Eventually you reach the entrance to the Royal Crypts.
Your new task is to find the Crypt of the Skeleton King. Explore the Royal Crypts. You will come across Leoric's Sword. Interact with it for a small story scene, showing the ghost of Lachdanan executing Leoric. Keep going, there is a room with four pillars. There are arrow traps in this room. The ghost of Leoric will activate some pillars that summon skeletons. Destroy the activated pillars, and the enemies. You should focus on one pillar at a time, and let the Templar deal with the spawns. If you have AOE attacks, use them to aid Kormac a bit. This is a good place to get a Massacre record, a record for number of monsters killed at once. Or you could try and make a run for it, but you may die several times in the process. Soon you reach an Iron Gate, which is the entrance to the Crypt of the Skeleton King. Go through, and your objective will change to: Enter the Crypt of the Skeleton King.
Go through a heavy door. You should get the achievement:
Thorough Inspection
Explore all areas of the cathedral in Act 1
There is a small cutscene. The Skeleton King collapses the bridge in front of you, and sends his minions to fight you. Go around, and kill the Returned on your way. Next, enter the main chamber and place Leoric's crown on the Skeleton King. Your task now is to kill the Skeleton King, Mad King of Tristram
Skeleton King Strategy Guide
The Skeleton King is the toughest opponent yet. But it is possible to kill him Solo by using this guide.
Skeleton King has a lot of health, and his attacks do devastating damage.
The Skeleton King uses the following attacks and tactics against you:
Mace Attack (Melee)
Summon Skeletons (Magic)
Teleport Attack (Magic)
The skeletons he summons are two Forgotten Soldiers, and two Returned. He casts two of those spells usually, resulting in an addition of eight undead foes to this already difficult fight. Use Area of Effect Attacks if you possess those, as those adds are a good source for health globes. Make sure to use the red globes only when you really need then. When you see him about to perform a whirlwind attack, move back a bit so that it misses you (and perhaps even kill his own soldiers). Stay on the move if possible, both to avoid the Skeleton King's melee attacks, and to stay clear of the summons. This fight might take a long time, and you will possibly die a few times in the process, reviving in the latest checkpoint and starting over. But follow the tips here, and you should be able to defeat the Skeleton King.
To summarize, the strategy for defeating Skeleton King is:
1) Be moving all the time, avoiding the Skeleton King and his minions. You can circle around the area.
2) Let your follower deal with the summons. The Templar does not die, he just is incapacitated when his health gets low. When possible, you can help him a bit with AOE attacks, or the occasional swing at a skeleton. But for most, your focus should be Skeleton King.
3) Strike using ranged attacks (if Demon Hunter, Witch Doctor, Wizard) and continue moving as he prepares his melee attack. Or if a Melee Class (Barbarian, Monk), use Hit and Run tactics, striking and retreating
4) Use the red health globes effectively. You should have plenty of supplies, don't go too close to a red globe if you do not need healing.
Kill the Skeleton King to advance the quest.