Very few people living in Hawaii today have only Hawaiian ancestry. Many are descendants of laborers who came to work in the sugar fields of the Hawaiian islands. The largest groups of immigrants to Hawaii came from China (starting in 1852), from Portugal (starting in 1878), from Japan (starting in 1884), from Korea (starting in 1903), and from the Philippines (starting in 1906). Today, those of Japanese descent constitute roughly a third of the population, the largest ethnic group in Hawaii. There was not a significant American population in Hawaii until 1875.
An easy way to track ancestors who migrated to Hawaii is through port records. Since the main Hawaiian port is Honolulu, it is easy to track your immigrant ancestors there. Check Bernice Judd's Voyages to Hawaii Before 1860 and the Hawaii State Archives, which maintain microfilmed port records as well as entry permits and labor permits.
Aside from microfilmed records, there are many government records for Hawaii that have been digitized online. The digitized collections include: Indexes to Marriage Records, 1826-1929; Indexes to Divorce Case Files, 1848-1915; Indexes to Probates, 1847-1917; Indexes to Wills, 1852-1916; Indexes to Citizenship Records and Naturalization, 1844-1894; Indexes to Citizenship Records and Denization,1846-1898; Indexes to Citizenship Records and Passports, 1845-1874. All of these records collections can be found online at:
http://archives1.dags.hawaii.gov/gsdl/cgi-bin/library. You can also view World War I service records there.
Don't overlook the Social Security Death Index as a good source for more recent, twentieth-century family members. You can find Hawaiian relatives in the Social Security Death Index for free on www.familysearch.org. USGenWeb also has a page devoted entirely to Hawaii and full of excellent links and resources. You can find this page at: http://www.higenweb.org/. It contains county histories and databases, immigration and passenger records, and much more.
One good resource for Hawaiian research is William A. Cole's Cole-Jensen Collection, which contains 51 books of oral genealogy transcripts collected from Hawaii. This collection is available on microfilm from the Family History Library.
Another good resource is Edith Kawelohea McKinzie's Hawaiian Genealogies: Extracted from Hawaiian Language Newspapers.
Good luck searching. Pomaika`i!