While The Dream Quest of Unknown Kadath does not come separated into chapters, from reading the story There is a definite structure and tone shift between parts that could easily be separated into chapters. I'll be covering each of these "chapters" in a separate comment chain.
The first "chapter" establishes the quest on which Randolph Carter is to embark, and the hubris of Carter. Carter has received three dreams of a fantastic city, only to wake before he is given a chance to explore. He prays for answers to the gods of dream who reside atop Unknown Kadath, but they do not seem to respond. He makes sacrifices in the cavern-temple holding the Pillar of Flame, which separates the 70 steps of light slumber from the 700 steps of deep slumber leading to the Dreamlands. Of note is that these 770 steps act to expert dreamers as a gateway from the waking world to the dreamlands. After his prayers and sacrifices, he receives no more dreams of the fantastic city. He thus resolves to seek out Unknown Kadath, the final home of Earth's gods, and beseech them in person to reveal this city to him.
Everywhere he goes, he is warned away from his quest by the wiser denizens of the dreamlands. The priests who tend the Pillar of Flame, Nasht and Kamon-Thah, state that the will of the gods has been communicated to Randolph Carter in that he has not received further visions. Indeed, it is unknown whether the gods reside on a peak located in Earth's dreamlands, or in a portion of the Dreamlands far away in the cosmos. Only three have ever dared the journey into the cosmos of the dreamlands, and of those three only Kuranes, king of Celephaïs, has returned at least partially sane.
Ignoring their warning, he descends to the Dreamlands and contacts the Zoogs, rodent creatures of the Enchanted Woods, who Carter notes have previously predated on lost dreamers travelling in those woods. Carter has made friends with these people before and has learned much of the Dreamlands from them. The Zoogs do not know the location of Unknown Kadath, but they escort him to the edge of the woods and towards the town of Ulthar, where a character from a previous story may have more information.
In Ulthar there is is the last extant copy of the Pnakotic Manuscripts, secreted away from the waking world into the Dreamlands before the destruction of Olathoë following the events of the short story Polaris. Another important source of information is the high priest Atal, The only living person known to have survived a journey to contact the gods of Earth. Atal has grown from an acolyte to a high priest over 300 years old, and in this time he has grown very wise. He speaks of his adventure and the loss of his mentor Barzai from the short story The Other Gods. He once again warns Carter against his quest, but Carter continues. Carter betrays the trust of Atal by getting him drunk, so that Atal will speak more freely. Atal drunkenly babbles of the mountain Ngranek in the distant south, where the likeness of the gods is carven into the most treacherous face.
Carter hatches a plan with this information. He will travel to Ngranek and witness the carving. It is known among dreamers that the younger gods enjoy disguising themselves, descending from their home, and mating with the women at the foot of their mountains. Thus, Carter will travel the Dreamlands and inspect the people who live there. Surely, the land where the people most resemble the face on Ngranek must bear the blood of the gods in the highest concentration. It is there that he will find Unknown Kadath.
He ventures to the port city of Dylath-Leen, where he enquires unto passage to Oriab, where the mountain Ngranek lies. While waiting for a boat which offers passage to oriab, a black sailed slaver ship docks in the city. The locals are made uneasy by this ship. the crew appear to dress in pointed turbans and tiny shoes. The rowers of the three banks of oars are never seen despite the long period for which the ship is docked. Ostensibly, no food is taken on board for crew or rowers, but slaves are paid for "by the pound." The slavers trade exclusively in gems, mainly rubies.
A merchant from aboard the slaver ship makes acquaintance with Carter, hinting at secret knowledge that he cannot divulge. Carter attempts the same trick as he performed on Atal, trying to get the merchant drunk. The merchant gulps down the wine, but doesn't appear phased. He then offers Carter a wine on a carved ruby carafe. After just one sip, Carter finds himself losing consciousness as the merchant laughs evilly.
This is what I consider the end of the first chapter of Unkown Kadath. Already it's filled to the brim with references to previous stories, giving much pay-off for our previous reading. Of note are The Cats of Ulthar, The Other Gods, Celephaïs, and Polaris.
We see a similar fate befall the Zoogs as did the cat killers in The Cats of Ulthar; while nothing explicit is mentioned yet, the Zoogs seem to have attempted to prey upon a cat only to have the Cats of Ulthar attack and sate themselves on the bodies of the Zoogs, who are missing when Carter next looks for them.
We know of a relative timeline in stories, as Atal, who was a young acolyte in The Other Gods, is now a high priest of over 300 years. Of course the timeline of the Dreamlands and the timeline of Earth seem mostly disconnected, but it's handy to have an idea of the continuity within dream stories.
We see mention of Kuranes, who Carter remarks as a friend who he knew by a different name on Earth. It makes sense that an avid dreamer such as Carter would seek out other dreamers in the waking world. It's a nice nod to Celephaïs.
Finally, we get some more information about the ancient city of Olathoë, which is the source of the much cited Pnakotic Manuscripts. Evidently the Dreamlands keeps more accurate ancient records (possibly aided by the altered passage of time) and mentions Olathoë as an ancient boreal city in the waking world which was sacked by an invading group called the Gnopkehs. A lot of authors around this time were inspired by the alternate history of Hyperborea. Robert E. Howard, who was a contemporary and friend of Lovecraft, is probably the best known fantasy writer of that period, writing his Conan stories in an alternate history of Earth called the Hyperborean Age.
In Carter we see a common theme repeated over and over in these dream tales: hubris. Our protagonists are often driven by a sense of morbid curiosity, or even defiance against the warnings of others. They seek these dangerous sights in some way to prove themselves some manner of master. As in earlier stories, we see Carter suffer for his hubris when he is captured by these slavers. Where will our story take us next?