Online:Long Live the King

Quick Walkthrough
#Deliver the letters to allies.
#Get a bottle of the King's favorite ale.
#Go to the castle overlook.
#Participate in the funeral.
#Talk to Bazrag.

Detailed Walkthrough

At the end of the last quest and the beginning of this one, Bazrag gave you some instructions: to deliver letters to some allies around town.  Six letters are to be delivered: these allies may be different depending on whether you spared or killed Talviah Aliaria in an earlier quest.  Bazrag also wants a bottle of Kurog's favourite ale, and reveals that the Orcish participation in the Daggerfall Covenant was a private agreement with Kurog, and the clan chiefs must meet to consider whether to continue it as a nation.

Four of the letters go to the same people: 
Mulzah has regained control of the Greedy Gut, and is back tending it.  Unwilling at first to have any more to do with you or Bazrag, and just wanting to be left alone, she is intrigued enough to find out that it is a proper funeral for Kurog that she decides to attend after all.
Priest Uugus, at the Temple of Ire, is preparing for the temple to be rededicated to Malacath, since the city (and the clan chiefs) are turning against the Trinimac cult following the deeds of the Vosh Rakh. However, he will be honoured to attend, and see the wisdom of Bazrag's decision: the city needs healing, not revenge.
Ulsha, Bazrag's shield-wife, currently standing outside the Museum of Orsimer Glories, has been looking out for the survivors of the attack on Fharun Stronghold, and is intrigued to receive her husband's letter.
Bizra, standing by the river near the eastern crossing, is someone you have not met before, but she and her partner (for whom she is waiting, and will turn up in a minute) are the two that Bazrag specially mentioned if you asked him. Both are old companions of Bazrag: Bizra's partner is none other than Bumnog the Brawler, last seen trying to punch harpies to death in Glenumbra and Rivenspire if the player has done those quests. These two count for a single invitation.

If you saved Talviah:

Talviah Aliaria, at Mazabakh's house (with its underground connection to the Greedy Gut) will be understandably pleased that you spared him earlier. He will be sad that you could not save High Priestess Solgra, but be pleased to accept the invitation.
Eveli Sharp-Arrow, at the Temple of Ire, will accept the sixth invitation, though more reluctantly, and she is still mourning Solgra.

If you sacrificed Talviah:

Flies-In-Wind, at Mazabakh's house, will still be mourning the death of Talviah, but will also agree to come. 
High Priestess Solgra, if you saved her life earlier, is still tending to Eveli Sharp-Arrow (who is still alive but badly injured, and in no condition to receive an invitation herself in this case), and would rather not attend, although she believes Bazrag's heart is in the right place - but then, she had believed that of Kurog too. 

Finally, you have to get a bottle of Kurog's favourite ale from The King's Cornerclub - something which the barmaid Roxanne is quite happy to supply for free, but wants to come to the funeral herself (this is automatically agreed). After which, the next step is to head to the palace, up the stairs, and out to its overlook balcony, to begin the funeral, in which Kurog goes to the pyre.  Two women flank him in death: to the left is Forge-Mother Alga, while to the right is High Priestess Solgra (if she died earlier) or Shield-Wife Razbela (if Solgra survived).  Chief Bazrag gives a fine and inspiring speech in Kurog's memory, talks about there being clans without chiefs, a city without a king, all needing guidance, while asking the player to light the two torches: then reveals that the surviving chiefs have asked him to take on the role of king of the Orsimer, and that he has cautiously agreed.  Thus, the Orc who was originally the strongest opponent of Kurog's plan (albeit by argument only) ends up being at the head of it, Chief Bazrag becomes King Bazrag, and vows to continue on with Kurog's dream (thus, presumably, perpetuating the Orcish part in the Daggerfall Covenant alliance) but avoid his ambition, promising always to consult with the clan chiefs, and offer help to them if wanted, but never force it on them.  

Back in the palace, if you have helped some others in quests, they will be present as well - Rigurt the Brash, Alinon the Alchemist and the Vanos siblings in particular, along with those who were at the funeral.  King Bazrag will be up on the balcony, accompanied by Eveli Sharp-Arrow. If Solgra is alive, she will be up there as well, revealing that Bazrag has kept her on as a royal adviser, even though she retains her faith in Trinimac and thus is no longer in charge of the Temple of Ire, since that is being rededicated to Malacath: and Eveli has forgiven you for your decision at Paragon's Rest, and been offered a position training Bazrag's archers.  If Solgra died, Eveli will be Bazrag's only companion on the balcony: she will still be mourning Solgra, and has decided to refuse Bazrag's offer and instead leave Orsinium.  Bazrag, meanwhile, thanks you for your deeds, and declares that although your service to the Orcs is ended, you will always be welcome in Orsinium.

Notes
Although Solgra, if she is still alive, initially declines the invitation, she will in fact turn up belatedly at the end of the funeral, as Bazrag leaves: as will Eveli, still clutching her injuries, but now clearly on the mend. However, neither can be interacted with at this time, until the player returns to the palace.
Flies-In-Wind will still be present at the funeral, whether or not she actually received the invitation personally (if Talviah was dead) or not (if Talviah was alive to receive an invitation of his own).
As you turn to leave the palace, after speaking to Bazrag for the final time, you may get a vision of the late Emperor Varen Aquilarios warning of future troubles to come, "when the gates to Sotha Sil's hidden city stand open". This is a reference to what was at the time a future planned DLC, the Clockwork City - although, in fact, in the release timeline (and presumably in whatever passes for the in-game timeline), Clockwork City ended up actually being a follow-up to the "Chapter 2" of Morrowind, rather than being the next sequential adventure in its own right, and the Daedric Cabal plotline actually comprises Morrowind, Clockwork City and Summerset in that order, and make most sense when played in that order.  Varen's appearance is regardless of whether you have in fact already completed Clockwork City (or indeed Morrowind, or Summerset).
Once Bazrag is crowned, if the player has not yet completed the Orsinium Patron achievement, future contributions towards the achievement only cost about 10% of the previous cost in gold.
The chiefs who survived the moot and are present at the funeral, will include the ones you helped earlier - Ramash of Clan Tumnosh, Abzug of Clan Morkul, and whichever of Laurig or Ushruka you put in charge of Clan Shatul. (If you are an Orc and took the leadership of the Shatul clan yourself, then nobody else is standing where the Shatul chieftain would be. Technically this means that *you* are Shatul's representative, but nobody ever references this in-game, here or anywhere.)
The sidequest A Cold Wind From the Mountain cannot be started until this quest (and nearly every other quest in Wrothgar, and the Wrothgar Relic Hunter achievement) is complete. This is one of the only instances in the entire game in which a sidequest actually requires completion of the zone's entire main quest.
 

Quest Stages