TW: Suicide
This is a great question though one that can be difficult to answer. Right after the revolution there wasn't a clear political structure to replace the colonial powers. Dessalines declared himself emperor but that only lasted from 1804-1806 when his rule was ended by assassination. A civil war erupted and things settled into the republic in the southern regions and Christophe's kingdom in the north. (Technically, Christophe originally proclaimed himself president but then in 1811 decided to upgrade to king, which also made forming a strong centralized governance easier.)
The Kingdom of Haiti ruled by King Henri Christophe only lasted from 1807-1820, though. It did not last long in part due to the embargoes from other countries including the United States that prevented any economic growth through trade, the ongoing tensions and violence with the south, and his own suicide. This makes assessing details like what the day-to-day of the royal court was like difficult at times and as you might imagine it could change rather quickly. I'll try to relay what archaeologists and historians know about the structure of the kingdom, though.
King Christophe broke up his kingdom into three administrative divisions and six districts with state officials in each to manage things like agriculture. "Code Henry" required incredibly intense and brutal agricultural outputs because despite the embargoes they managed to find work-arounds to still engage in at least some trade. And that trade was vital for any hope that he could turn the kingdom into something economically stable, powerful, and successful. He did this by recreating the European model of the feudal system including inventing heraldry. These heraldic patterns adopted by his Royal Court were “calquées” from French heraldic models if you're curious. These included mythic European creatures mixed with local Caribbean animals like guinea fowl. He also created official titles and positions like barons, the maison du roi, dukes, and even an official librarian (who had his own arms that were books on a bookshelf.) The librarian is particularly interesting because given that most of the populace were formerly enslaved most were illiterate. All had responsibilities over various sectors and parts of governance.
In addition to being an existing model for how to construct a centralized and organized governance, the goal was to prevent re-invasion from Europeans by proving that Haiti was equal to those European institutions. And this attempt wasn't entirely unique - Mexico and Brazil both adopted independent monarchies that were also modeled on European forms in the 1820s. But all three were creolized, meaning that they also incorporated local forms of power relations and calls to pre-colonial symbols of power. Christophe had a standing army that was modeled after Dahomey with the Royal Dahomets as well as the Society of Amazons (women soldiers) also modeled after the Dahomey. This hybrid mix of symbols and structure allowed him to appeal to the mixed reality of his populace. There were the newly minted nobility who largely came from pre-existing free people of color who had more power in the old colonial system as well as those who rose up socially through the revolution and loyalty to Christophe. They were exposed to European symbols and structures as the way to set up a system of power. And then there were the rural largely agricultural communities still deeply connected to West African power structures, which were also monarchical and for whom these Dahomey symbols were more meaningful. The buildings he constructed had similar dual impacts.
Christophe built an impressive amount during his short reign: nine royal palaces and fifteen fortresses. He also renovated a lot of plantations to turn into seats of power and governance. Sans Souci (which you can still tour today and I highly recommend it!) was his primary seat of power and covered 13 hectares. Quoting Monroe, "The complex contained numerous additional buildings: individual residences for Christophe, the queen, and the heir apparent; a military garrison for his Royal Dahomets,residences for state officials, and two extensive gardens. The ground plan and façade of the main palace reveal strict adherence to aforementioned neoclassical principles of separation, bilateral symmetry, the control of movement through space, and the use of Greco-Roman elements. These elements earned Sans-Souci “the reputation of having been one of the most magnificent edifices of the West Indies”(Brown 1837:186),described as a veritable Versailles in the Caribbean because of its splendor" (235). Reports vary as to when it was completed but by 1809 reports suggest he was living there. The neoclassical architecture signaled he was modern and "civilized" while the massive structures also highlighted his power to locals.
Inside it was decorated to highlight that the Kingdom of Haiti was just as refined and beautiful and powerful as any European power. "The whole first floor contained a great many halls, richly decorated according to European taste. We were astonished at the devastation here. Not infrequently, we had to step over beautiful draperies or debris from mirrors. The furniture was made of mahogany wood, the windows covered with silk curtains and the floors polished. I saw glass windows here for the first time in this country, I even encountered a few glass paintings [Glasmalereien] in the apartments. (Ritter[1836:77–79], cited and translated in Bailey [2017:88–89])" (cited in Monroe pg 236). Paintings of Christophe had him dressed like a European royal and the English confiscated a ship full of regalia headed to Christophe that included bejeweled crowns and chalices.
Monroe points out that mobilizing very English (rather than French) forms of nobility and material culture may have been strategic to try and get the English to recognize and support the fledgling nation. This is supported by McIntosh and Pierrot who show that Christophe would modify presentations, language, and depictions of himself to appeal to English interests (he was also fluent in English.) England was certainly intrigued and you can read about some of these paintings and communications here. It also includes some images of the portraits if you're curious.
In 1820, Christophe suffered a stroke that left him partially paralyzed. News of his infirmity led to unrest and threats to his power. On October 8th, 1820 he had his attendants bathe and dress him in his military finest and once alone he shot himself. Ten days later his only surviving son and heir apparent was murdered. The kingdom was no more.