The earliest known census was taken in 1843. But, the earliest known census with recorded names begins in 1878 for the islands of Oʻahu, Maui, and Hawai'i, 1890 for all the islands, and 1896 just for Honolulu. After Hawaiʻi became a territory of the United States, the first U.S. census in Hawaiʻi began in 1900, and happened every 10 years since. Census usually contains lists of all persons in household, sex, age, marital status, citizenship/nationality, occupation, and ownership of livestock and property. The census are organized by island, then by district. The first thing you want to do is generate a list of your ancestors that you think would be alive during a certain census, then think of where they would be located. For the early censuses (before 1900), you may be able to find ancestors through FamilySearch. If that does not work, you will need to manually search through each census: page by page, line by line, family by family using the the digital or microfilm versions. For the later censuses (after 1900), you can do a keyword search for your ancestors online in FamilySearch. More information about how to access these various census records is listed below.
Census in the kingdom was taken by the government. Originals are located at the Hawaiʻi State Archives. Currently, these census are available online through the new Hawaiʻi State Archives Digital Archives Catalog, on microfilm at Hamilton Library and in print at the Hawaiʻi State Archives.
You can now search names in some of these census records through FamilySearch's collection, Hawaii Island Census Records, 1878-1896, which appears to be a compilation of the various census records held at the Hawaiʻi State Archives from 1878-1896 for some islands.
Note FamilySearch's collection is partial and not complete. If you are not finding any results, manually search through the original sources listed below. When those links are clicked, you will be taken to the record in UHM Library's OneSearch for the microfilm copy. You can also access these census records online in the Census Collection through the new Digital Archives Catalog.
Territory of Hawaiʻi
When Hawaiʻi became a "territory" of the United States, the decennial census (years ending in "0") were taken every 10 years beginning in 1900. Each census is released for research 72 years after it was completed. So far, the 1900, 1910, 1920, 1930, and 1940 census are available on Ancestry.com (public library subscription or paid), FamilySearch (free), HeritageQuest Online (public library subscription).