You might have to search several fake defiled crypts until locating the Chancellor's Altar. Your character will say something like "The crown is not here. I will seek it in another crypt" when you have explored the whole crypt, as a hint that lets you know to move on to the next crypt. As mentioned before, it is a good idea to search all the crypts anyway, for valuable items, events, side quests, achievements, and experience. Remember to always use Enlightened Shrines when you run across them. Those will give you a blessing that will boost your experience gain by 25%.
The crypt containing the crown will be bigger than the other crypts. It will be full of Skeletons and Carrion Bats, as well as Ghostly Murderers. Venturing through the corridors, you will eventually reach the exit to the Defiled Crypt Level 2. Your character will say something like "The bones beneath this tomb scream of madness and betrayal. The crown is near" (these dialogs often differ depending on the class you are playing).
You arrive at a big door. Open it, and reveal a boss monster: Manglemaw, Horror of the Crypt. He is a huge boss, with lots of hitpoints and devastating melee attacks, so plan your strategy accordingly. If a ranged character, keep moving and shooting him from a distance. A character such as a Witch Doctor can send minions to attack Manglemaw from melee range, drawing his focus, while the witch doctor himself shoots the boss from afar.
After defeating Manglemaw, enter the Chancellor's Tomb. If available, use the protection shrine there to receive a blessing that will reduce damage taken by 25%. Fight your way through the skeletons and undead tomb guardians. Reach a big chamber with the Chancellor's Altar in it and interact with the Altar. The ghost of Chancellor Eamon appears. He will not allow you to bring that monster Leoric back to this world. Your task now is to kill Chancellor Eamon (Astral Guardian). Defeat Eamon's Ghost, then you need to take the Skeleton King's crown from the Chancellor's Altar. But you see that Leoric's crown is broken, and you hope that Haedrig will be able to mend it. You now need to use the Town Portal to return to New Tristram.
In one of the corridors nearby, there is a portal, Use it to travel from the Chancellor's Tomb to new Tristram. Then, talk to Artisan Haedrig Eamon who is standing near the town portal. He did not think you would make it back. He recognizes that the crown need repairs, but tells you it should be no problem. He fixes it, then tells you it is new as the day it was placed upon the Black King's Head.
This completes the quest, A Shattered Crown.