How and when did the U.S. begin its influence in Hawaii? How was Hawaii annexed?

The United States started down the road to annexing Hawaii in 1820, when the first missionaries arrived in the islands from New England and set up shop on Maui, Oahu and the big island. At the same time, whaling ships from Massachusetts began stopping at the ports of Honolulu and Lahaina to stock up on provisions.
Their arrival coincided with the erosion of traditional Hawaiian customs and beliefs following the death of the great king Kamehameha the previous year. The power struggle between the new king and the dead king's favorite wife precipitated crises that hastened the abandonment of centuries-old traditions.
As the Hawaiian islands opened to the West, the native peoples enjoyed trading with the newcomers. This trade was boosted by the opening of large commercial sugar and pineapple plantations in the 1830s and 1840s, and the California Gold Rush made Hawaii an attractive trading partner due to its proximity to the West Coast compared to the US Eastern Seaboard. Western influence increased with the 1840 Constitution and an 1848 land act that allowed foreigners to gobble up lands previously owned by nobles and commoners alike.
Hawaii signed a most favored nation treaty with the US in 1849 that provided generous terms to Hawaiian sugar producers -- a situation that lasted until 1890, when tariff rates on foreign sugar imports hurt the Hawaiian economy greatly. The growers, mostly wealthy Americans, saw annexation as a way to make the tariff problem disappear. They started an uprising in 1893 that forced Queen Liliuokalani to abdicate, and an annexation treaty was quickly drawn up, but before it could be approved, newly inaugurated US President Grover Cleveland withdrew the treaty from Senate consideration.
President Cleveland believed the US had acted wrongly in Hawaii and was determined to place the queen back on her throne, but his attempts to do so were thwarted at every turn. Sanford Dole, who'd led the uprising that overthrew the queen, refused to step aside and declared Hawaii an independent republic. When William McKinley became president in 1897, he negotiated a treaty with the Republic of Hawaii, and when war with Spain broke out a year later, the naval base at Pearl Harbor and its strategic location for the war effort sealed Hawaii's fate. Congress quickly approved formal annexation of the islands, and by 1900, Hawaii was organized as a U.S. territory, which it remained until statehood in 1959.
https://www.history.com/this-day-in-history/americans-overthrow-hawaiian-monarchy