Skyrim:A Blade in the Dark

Quick Walkthrough
# Talk to Delphine.
# Locate the dragon burial site.
# Kill the dragon Sahloknir.
# Talk to Delphine.

Detailed Walkthrough
A Harmless Innkeeper
Now that you know that the Horn of Jurgen Windcaller has been pilfered from Ustengrav, follow the instructions in the mysterious note left in its place. This will lead you to Riverwood and the Sleeping Giant Inn. Speak with the innkeeper, Delphine, about renting the attic room. She will tell you that there is no such room, but you can take the room on the left. Delphine will follow you to the sparse room she indicated, where she will address you as Dragonborn and give you the Horn. She cryptically tells you that you and she need to talk and asks you to follow her.

The Problem with Dragons
Delphine will lead you to a room across the inn. She will ask you to close the door to the room before both you and she can proceed any further. Do it, and she will open a secret passage in the back of a wardrobe that leads down to a subterranean room.

Here Delphine will reveal that she took the Horn to attract the attention of the Dragonborn. The note was a safeguard to protect against trickery by the Thalmor. She also tells you that the dragons are not just coming back, but are actually coming back to life, resurrected from their burial mounds. Only the Dragonborn can permanently kill dragons by absorbing their souls. But she is not quite ready to accept you as Dragonborn, despite the Greybeards' interest in you. Luckily, she has a task to prove your authenticity. According to the Dragonstone map taken from Bleak Falls Barrow, the next dragon will rise from a burial mound in Kynesgrove. She wants you to accompany her to the burial site and kill the dragon; only then will she accept you as Dragonborn.

If you tell her you're ready to leave, she will put on her gear and leave the inn. You are free to follow Delphine to Kynesgrove, and she will briefly talk to you at waypoints about her concerns. Or you can go at your own pace and possibly beat her there. 

Kynesgrove
As soon as you get to Kynesgrove, Iddra, the innkeeper, will run up to you and tell you that a dragon was sighted flying over the burial mound and that you should get out of town. If you traveled with Delphine, she will run up ahead of you to the burial mound, fearing it may be too late. If you arrived ahead of her, she will appear shortly thereafter on the mound behind you.

Speaking with Dragons
Upon arriving at the site, you and Delphine will witness the dragon, Alduin, using his Thu'um to resurrect a dead dragon. Alduin cannot be killed or injured at this time. When the dragon, Sahloknir, is whole once more, he has a conversation in the dragon language with Alduin.

Alduin: Sahloknir, ziil gro dovah ulse! 
Sahloknir, ever-bound dragon spirit! (spirit bound dragon eternity-of.)
Alduin: Slen tiid vo!
Let your flesh be restored! (Flesh against time.)
Sahloknir: Alduin, thuri! Boaan tiid vokriiha suleyksejun kruziik?
Alduin, my overlord! Has time come to revive the ancient realm? (Alduin, my overlord! have flown time to revive realm ancient?)
Alduin: Geh, Sahloknir, kaali mir.
Yes, Sahloknir, my trusted ally. (Yes, Sahloknir, my champion [of] allegiance.)

Here, Alduin attempts to speak to you, addressing you as Dovahkiin.
Alduin: ''Ful, losei Dovahkiin? Zu'u koraav nid nol dov do hi.''
So, you're the Dragonborn? I do not recognize you as dragon. (So, you are Dragonborn? I recognize no from dragon of you.)
Alduin: You do not even know our tongue, do you? Such arrogance, to dare take for yourself the name of Dovah.
Alduin: Sahloknir, krii daar joorre.
Sahloknir, kill these mortals. (Sahloknir, kill these mortals.)

After ordering Sahloknir to kill you, Alduin flies away. Delphine will help you kill Sahloknir, and upon his death, when you absorb his soul, Delphine will be quick to acknowledge you as Dragonborn.

Paranoia
Speak to her, and she will reveal that she is one of the last members of the Blades. She will also tell you about her suspicions that the Thalmor are responsible for the return of the dragons. Since the Thalmor are part of the Aldmeri Dominion and nearly destroyed the Empire during the Great War only 30 years ago, they have a motive for rescuing Ulfric in Helgen, simply in order to keep the war raging. The Thalmor stand to benefit most from damage to the Empire by the dragons.

Finally, she will reveal her plan to infiltrate the Thalmor Embassy to search for evidence of their connection with the dragons. She has no idea how to get past embassy security, and asks you to meet her back in Riverwood. This quest concludes, and the Diplomatic Immunity quest starts.

Notes
While A Blade in the Dark is active, dragon locations will be limited to dragon walls. Other than dragon walls, random dragon encounters are eliminated while A Blade in the Dark is active allowing for controlled access to dragon fights.
Despite it saying you need gold to rent the "attic room" (and Delphine taking said gold if you do), you do not actually need the gold on you to progress the quest.
When Delphine goes to put her traveling gear on at the start of the quest in Riverwood, she will stand next to the chest, although the chest beforehand states that it is empty.
Delphine will not acknowledge your status as Dragonborn if you slay a dragon that is not Sahloknir, despite any soul absorption.
You can happen upon the burial site any time after Delphine leaves Riverwood. The quest will progress once you arrive.
Standing too close to the burial mound when Alduin revives Sahloknir may cause you to be thrown aside in a manner similar to the Unrelenting Force shout. Though you can likely pick yourself back up before Alduin's and Sahloknir's conversation ends and the battle begins, you can be thrown off the hill and take fall damage, so be careful.

Bugs
 
Make sure you have shut the door behind you before talking to Delphine - she asks you to do this. 
 This can be fixed by typing   in the console, depositing the Horn of Jurgen Windcaller in your inventory and skipping to Delphine's 'we need to talk' dialogue.
 
 
This can be fixed by walking far enough out of the dragon's range, which extends as far as Kynesgrove, and walking back.
 
This can be fixed by reloading a save and before going to Kynesgrove, make sure to first complete the "Horn of Jurgen Windcaller" Quest by going to the Grey Beards and returning the Horn to them. Also, make sure that you have learned the shout "Become Ethereal" in Ustengrav, and activated the altar where you found the note from Delphine to gain a Dragon Soul. Use that Dragon Soul to activate "Become Ethereal". Then go meet Delphine in Kynesgrove and slay the Dragon. 
Uninstall/Disable the mod "View Yourself VR" if you have it, because it has been reported to cause this bug.
 Using the console command   can correct this so that the player will progress to the point where Delphine will offer to tell all.
 Using the console command   can correct this. Follow up with  .
 If Delphine is still not responding, the main questline can be forwarded to the beginning of the Diplomatic Immunity quest by using   followed by  .  This will cause Delphine to run back to the Sleeping Giant Inn in Riverwood, where her dialogue will be reset for the new quest.  It will also add Delphine's Secret Door Key to your inventory.
 If this also doesn't work, you can load your last saved game, slay Sahloknir, absorb the soul and then move very slowly to Delphine. Don't talk to her, wait until she finished all her lines and repeats the last one - then talk to her.
 
 
 
 

Quest Stages