Tribunal:Hunt the Dark Brotherhood

Quick Walkthrough
#Enter the Manor District of Old Mournhold, from the Bazaar Sewers.
#Enter the North Building of Moril Manor and kill Dandras Vules, the leader of the Brotherhood.
#Search Dandras' corpse to find a clue about what's happening.
Alternatively pickpocket the contract from Dandras. Killing him isn't mandatory.

Detailed Walkthrough
Mournhold Palace
In the opening quest, you spoke with Asciene Rane about Transport to Mournhold, and arrived in the Royal Palace, teleported next to the Argonian Effe-Tei, who can teleport you back to Vvardenfell. Following Asciene's advice, you spoke to any Royal Guard or High Ordinator about the Dark Brotherhood. You were directed to the sewers underneath the Great Bazaar in the eastern section of town. 

That closed the first Dark Brotherhood Quest, and you might have done so as early as level 1 (6 in the Xbox version).  This second DB quest is much more difficult.

 

Finding the Great Bazaar
Exit to the Royal Palace Courtyard from the door next to Effe-Tei and from there continue to Plaza Brindisi Dorom. Head left, following the wall, to find a doorway leading to the Great Bazaar. The entrance to the Bazaar Sewers is in the southeast corner of the Great Bazaar. (Mournhold has various sewers; this is the one you want.)

Finding the Dark Brotherhood
Explore the Bazaar Sewers, watching out for undead, Goblins, and Durzogs that live down here, until you find one of the two entrances to Old Mournhold's Manor District ruins.  Caution: Some of the Goblins are armed with the Goblin Club, one of the highest-base-damage weapons in the game, and can kill even many level 20ish PCs in one to two hits. Many also carry and use healing potions, and they have resistance to either fire or normal weapons, depending on Goblin type.

As soon as you enter the Manor District, you will receive a Journal entry mentioning that you have found the Brotherhood's hideout (technically, this is the actual start of the quest). It won't be long until you find some masked Dark Brotherhood Assassins who would like nothing better than to kill you, and you'll encounter more the further you explore the area.  They are typically armed with poisonous and/or paralyzing weapons.  Aside from their boss, the Assassins respawn like generic Guards, and will continue to do so after his death.  They are leveled opponents, so you will not encounter every variety when you are low level.

Dandras Vules
In the northern building of Moril Manor, you will find boss Dandras Vules (he's the one without the mask helm). He is a tough opponent (level 55, and over 400 health), with a bow as well as melee weapons, so ensure that you are fully prepared before you take him on. Once you kill him, you will find an interesting Dark Brotherhood Contract on his corpse. You receive another Journal entry, which suggests you go talk to another Guard to find out more about the contract. Any Royal Guard will direct you to speak with Tienius Delitian, while any High Ordinator should direct you to speak with Fedris Hler, the Chief Steward of Almalexia in the Temple Reception Area.
You can also sneak past the assassins and steal the contract from Dandras. The game doesn't care which way you obtain it, but killing Dandras isn't required to advance the quest.

The Dark Brotherhood Contract
If you read the document, signed by "H.", you'll come to the conclusion that the client who hired the Dark Brotherhood to assassinate you is King Hlaalu Helseth himself (a new Journal entry will come to this conclusion, even if you did not as a player). If you go to Delitian, and give him the contract, he'll tell you frankly that it was Helseth who wanted to kill you, and he refuses to pay any more attention to that little matter.

Next Quest Options
Delitian will now have an optional series of six side quests for the Royal Palace which will eventually lead you to Hler for the next phase of the Main Quests. You can choose to skip the Royal Palace quests and go to Hler directly. If you went to Hler first, you can go to Delitian for the side quests later, as long as Barilzar's Mazed Band isn't completed yet.

Notes
It is actually possible to skip closing the first Dark Brotherhood quest: If you do not speak to a Royal Guard or High Ordinator about the Dark Brotherhood, that quest will remain open, and this one will not open when you find the Brotherhood's lair on your own. Instead, it will open when you kill their boss. You can still close the first DB quest by later talking to one of those two Guard types about the Brotherhood. 
This quest, and any of Delitian's up to the Queen Barenziah quest, are optional if you speak to Fedris Hler right away when you arrive in Mournhold.
Delitian will dismiss any attempt at dialogue with a chuckle, until you have completed this quest.
For Dark Brotherhood ranks, see Dark Brotherhood; for DB Assassin stats and the player levels at which each variant appears, see Dark Brotherhood Assassins.
The Dark Brotherhood base is arguably the best source of continuous income in the entire game, as the respawning Assassins there provide you with an unlimited supply of good gear to sell. Mournhold provides nearby vendors, many with larger amounts of money than most in Vvardenfell, and often with high-end items you may wish to trade for without draining your extant bankroll. The Dark Brotherhood Armor is very light but relatively valuable, and the higher Assassin ranks also wield a Daedric Tanto, Daedric Wakizashi, or Adamantium Jinkblade of Wounds.
The Alteration skill book Sithis can be found on a ledge outside the door to Moril Manor, North Building.
Though this quest is recommended for level 25+, it can be started much sooner, especially if some techniques (e.g. Invisibility, strong Chameleon, or Calm Creature) are employed to avoid death-by-Goblin, and one also avoids taking on Dandras Vules or too many Assassins at once. The Assassins are leveled opponents, so you should be able to handle them one at a time. You can also hire the mercenary Calvus Horatius and give him good gear, to increase your chances, though you're fairly likely to get him killed.  If you can get a Goblin Club for him, this will even the odds a bit.  He's not a spellcaster, so be prepared to Cure Poison on him after or even during Assassin fights.

 

Bugs
 
A "brute force" way to prevent this is to avoid fighting him and use the Console to progress through the quest. If this does not appeal to you, simply savegame before entering Moril Manor and reload your save if this bug (or another of those documented here) is triggered.
 

This bug is fixed by the Bloodmoon patch and the GotY edition.
 
 
Workaround: Use   in the Console to toggle off collision, enter the wall to access his corpse, back out of the wall, and toggle collision on again.
 

Quest Stages