Skeleton King, also known as Leoric is a raid boss in Diablo III. You will have to kill him, but first you need to complete a few quests. The quests you need to complete in order to be able to kill Skeleton King are: The fallen star, The Legacy of Cain and A Shattered crown. After you finished this 3 quests, you will unlock a new quest: Reign of the Black King. This is the quest in which you are requested to kill the Skeleton King.
If you don't know how to kill skeleton king in Diablo 3, follow the steps below and your success rate will increase significantly. Before you can actually kill the Skeleton King, you will have to place Leoric's crown on the Skeleton's head. To do that, just click on the raid boss. This is how it looks before placing the crown on his head.
Now, depending on your class you have several things to do so you can kill skeleton king. If you are ranged class: demon hunter / wizard then you will have to keep the distance from the raid boss because you can't handle too much damage. Also, take note that the Skeleton King will spawn from time to time some mobs that will hit you. Kill the mobs first because they'll drop health potions that you might need. After the mobs are dead keep range and still hit the raid boss. It will take you some time but it's the safest way.
In case your class is monk or barbarian then you can't really hit from range so you'll have to manage your hp wisely. Kill fast the mobs that the Skeleton King spawns and when your hp drops low take the potions from the ground.