Activating the quest
Quest requirements: This quest can be completed at any time during the game. However as it involves a series of fights with the strongest kinds of Draugrs, it would be good to approach in the further part of the game.
Start off by heading to Markarth, located in the western part of the map (screen above). If you haven't been there yet, it can be easily reached by using a carriage (e.g. the one stationed in Whiterun Stables).
After reaching Markarth, you can move to activating the quest at once or spend some time on learning of it officially. If you choose the latter option, I'd recommend heading to Silver-Blood Inn located right beside the city entrance. After getting inside, find Kleppr the innkeeper and ask him for the newest gossip (screen above). He will tell you that the Hall of the Dead has been closed for unknown reason, therefore unlocking Miscellaneous: Speak to Verulus about the Hall of the Dead.
In order to meet with Brother Verulus, you will first need to reach the Understone Keep. The entrance to this monumental construction can be found in the western part of Markarth (screen above).
After getting inside, look around for Verulus. If it's your first visit, he should be talking with Thongvor (screen above), demanding the access to the Hall of the Dead as fast as possible. if you were here before, the mage won't be talking with anyone and might be standing nearby the entrance or in the southern room.
Approach Verulus and speak to him to learn that the Hall of the Dead is temporarily off-limits. You will be able to choose from a few methods of solving this problem (screen above). If your character has highly developed Speech, you can try and persuade him to change his mind or intimidate. If those methods fail, you might also consider bribery. If you manage to convince Verulus into letting you inside the Hall of the Dead, you will receive the Markarth Hall of the Dead Key and unlock Miscellaneous: Investigate the Hall of the Dead.
Talking with Verulus fortunately isn't the only way of getting to the Hall of the Dead. You might try to pickpocket him, though it might prove quite hard considering he's often standing beside guards. You can as well try to break into the Hall of the Dead. For that, you can get out of the Keep and look for a proper gate or stay inside and head south-east (screen above). Also remember that no one can see you as you try to open the door.
Regardless of the chosen tactic (getting the key from Verulus, stealing the key, using lockpicks to open the passage), you will have to use one of the doors leading to the Hall of the Dead (screen above). Start exploring the area and from time to time you will hear the voice of a mysterious character speaking to you.
The mentioned character will soon reveal itself, regardless of whether you decide to take a look around the Hall of the Dead or stay at one place. Start the proper conversation with Eola (screen above), from which you will learn that she's a cannibal. Whether you remain neutral or be aggressive towards her doesn't change anything, as one way or another you will learn of the Daedric Lady of Decay and be invited to the Reachcliff Cave.
Heading to the Reachcliff Cave
New main objective: Clear the Draugr from Reachcliff Cave
Note that apart from the main objective, you also received a lesser one, namely Miscellaneous: Tell Verulus the Hall of the Dead is safe. I'd therefore suggest heading back to the main part of the Understone Keep. Find Brother Verulus and tell him that the Hall of the Dead is now safe (screen above). In return you will receive the Amulet of Arkay.
Head out of the Understone Keep and open the world map. Your destination is of course Reachcliff Cave (screen above). It's located south-east of Markarth and that's where you should begin your journey, unless you had discovered another location closer to the cave.
It's best to approach the cave from the north-east side, as there you will find a path leading into the mountains (screen above). Turn east at first chance to start going towards the cave in a straight line. On your way be wary of wild animals and Bandits.
After getting to the cave, you should note Eola standing before it. Approach the woman and initiate a conversation with her (screen above). You will need to decide whether you want to invite Eola to clear the cave of Draugrs with you or tell her to stay behind. The advantage of taking her with you is the fact that she will join in as an independent ally, so you won't have to give up on your current party member and will be able to explore the cave with two allies. The downside is that if she dies you will automatically fail the quest. Personally I'd recommend inviting her, though you will have to keep an eye on her and save more than usually.
Head towards the main Reachcliff Cave entrance. Save your game and go south. You will come across monsters in the very first bigger room and throughout the whole location you will have to fight the most powerful types of Draugrs. It's best to start this battle by attacking the Draugr Deathlord before he can get out of his grave and join the fight (screen above).
Further into the battle, I'd recommend attacking mostly the Draugr Scourge, who will most probably play the role of an archer (screen above). Only after the Scourge is dead should you focus on the Deathlord. Of course look out for his powerful attacks, dodging which will be especially important when playing as a character preferring ranged attacks and light armor.
Save your game once again and head into the corridor leading east. Be ready for a fight with new Draugr Scourges. Before the fight itself, you should consider increasing your ice resistance, as at least one monster will be casting such (screen above). Also remember to attack the Draugrs in order, especially considering that they're very resistant.