The Imperial Dynasty of America refers to two different things - a manuscript and an actual dynasty associated with Carcosa. Whilst this version of the document appears in the alternate 1920s (or just in the delusions of Hildred Castaigne), an updated version appears, some 75 years or so later, in a more familiar New York (as part of the Delta Green RPG scenario Night Floors).
As described in The Repairer of Reputations, the manuscript, with pages well-worn only from the constant handling and reading by Hildred Castaigne, concerns itself with the imperial dynasty that spreads from Carcosa to America.
The manuscript begins with the words "When from Carcosa, the Hyades, Hastur and Aldebaran" - indicating that each is an ancestral home of the dynasty, and continuing onto "Louis Castaigne, born December 19, 1887... Hildred Castaigne, only son of Hildred Castaigne Snr and Edythe Landes Castaigne, first in succession..." It is perhaps relevant that it never states explicitly that this is the end of the manuscript.
Mr. Wilde runs through the document late in the story for the benefit of Vance, to which Hildred Castaigne is witness:
"Mr. Wilde explained the manuscript, using several volumes on Heraldry to substantiate the result of his researches. He mentioned the establishment of the Dynasty in Carcosa, the lakes which connected Hastur, Aldebaran and The Mystery of the Hyades. He spoke of Cassilda and Camilla, and sounded the cloudy depths of Demhe and The Lake of Hali. "The scalloped tatters of the King in Yellow must hide Yhtill forever..." he muttered, but I do not believe Vance heard him. Then by degrees he led Vance along the ramifications of The Imperial Family to Uoht and Thale, from Naotalba and The Phantom of Truth to Aldones, and then, tossing aside his manuscript and notes he began the wonderful story of The Last King. Fascinated and thrilled, I watched him. He threw up his head, his long arms were stretched out in a magnificent gesture of pride and power, and his eyes blazed deep in their sockets like two emeralds. Vance listened, stupefied. As for me, when at last Mr. Wilde had finished, and, pointing to me, cried, "The cousin of the King," my head swam with excitement."
About the manuscript
It is not clear where this manuscript originates from, though presumably either Mr. Wilde or Hildred Castaigne penned it - given Mr. Wilde's familiarity with heraldry it would suggest the former. Although the document is only worn by Castaigne's hands Mr. Wilde seems to know the document well. In the passage above it's worth pointing out that Mr. Wilde tosses aside the manuscript and additional notes though before begin to relate the story of the Last King, suggesting that on some level there was a great deal of research put into the document. The story of the Last King, presumably, is something more prophetic than what seems to be a list of names and dates.
Claiming the crown
The majority of The Repairer of Reputations sees Castaigne planning towards ending his cousin's claim to the title of Last King and preparing himself for the position. It seems that Wilde and Hildred intend to stage a coup and take over both the Imperial Dynasty and then the USA. Wilde claims that they have 10,000 immediate supporters in New York, each of whom has received the Yellow Sign, with potentially millions more likely to back them, with only California and the North West states lacking in support for them. The entire coup idea would appear to be a figment of their joint madness, though whether Mr. Wilde has enough sway through his network is questionable.
As Castaigne himself relates, "I explained to Vance why I alone was worthy of the crown, and why my cousin must be exiled or die. I made him understand that my cousin must never marry, even after renouncing all his claims, and how that, least of all, he should marry the daughter of the Marquis of Avonshire and bring England into the question. I showed him a list of thousands of names which Mr. Wilde had drawn up; every man whose name was there had received the Yellow Sign, which no living human being dared disregard. The city, the State, the whole land, were ready to rise and tremble before the Pallid Mask... The time had come, the people should know the son of Hastur, and the whole world bow to the black stars which hang in the sky over Carcosa."
It's never quite clear why Mr. Wilde is prepared to do this for Castaigne, as he is seen acting as a monster towards others who have been touched by The King In Yellow. Somehow their madnesses seem to be in synch.
Upon the apparent success of his schemes, Castaigne explains: "At last I was King, King in my right in Hastur, King because I knew the mystery of the Hyades, and my mind had sounded the depths of the Lake of Hali. I was King! The first gray pencillings of dawn would raise a tempest which would shake two hemispheres."
Upon the realisation that he has failed, he watches his world fall apart: "I stood transfixed with rage and despair, seeing my crown, my empire, every hope and every ambition, my very life, lying prostrate there with the dead master."
There is the implication that, whilst Castaigne was happy to serve The King In Yellow, he was also serving Mr. Wilde in some capacity. Mr. Wilde would have been very much the power behind the throne.
1990s New York - Night Floors
There is a volume of the same name owned by (and possibly written by) Henri de Calvados Castaigne in the scenario Night Floors in Delta Green: Countdown. It consists of a leather-bound folio of loose and yellowed pages containing a complex lineage tracing a royal bloodline from Carcosa to New York . Over 1000 names are contained in it and on the final page, emblazoned in a splotch of red wax, is The Yellow Sign (the symbol).
Although over a 1000 names allegedly appear in the document, we only ever learn of a few members of the Dynasty itself, members of the Castaigne Family. The Dynasty originated in Carcosa, but the reason for terming it the Imperial Dynasty of America (beyond the coup plot) is unclear.
An unnamed member of the Dynasty's thoughts feature in the poem My Reputation Repaired, in which he seeks to claim his throne on the basis of a DNA report that may just be a blank sheet of paper.