He was born in Jamaica to a prominent plantation family, and was educated toward the law. Moore was active in Jamaica's colonial affairs, and by 1756 he had rise to the rank of Lieutenant Governor. As in many royal colonies of the time, the governor was frequently absent, collecting his fees and salary from London, with a local Lieutenant and council forming the actual government. Moore gained considerable reputation for leadership by suppressing a slave revolt during his first year on office. His reward was first to be made a Baronet, and then in 1764 he was named royal governor for New York.
He arrived in New York City with his family in November 1765. Relations between the colonies and England were very strained by this time, but not yet in open rebellion. New York City had seen riots and protests over the Stamp Act. The new governor calmed these by meeting directly with Isaac Sears, a leader of the Sons of Liberty. Moore agreed with Sears and the colony's assembly to suppress the Stamp Act, and gained additional goodwill be reducing the military fortifications within the city. His openness and courtesy earned him floral tributes while other colonial governors were being burned in effigy.
However, during the next few years, he actively used military force to suppress rural riots by tenants of the large estate owners. He ordered General Thomas Gage to actively pursue and suppress this form of rebellion. This did not seem to bring him any increased difficulty in governing, for two reasons. The first was that the Sons of Liberty also feared the introduction of rural problems into the city, believing that they should be the only ones to use riots as a bargaining tactic. The second was that the Assembly at the time was dominated by the Patrons, or large estate owners. In December of 1767 he even dissolved the Assembly to allow the Patrons to make up through new elections some of the numbers they had lost earlier.
He died suddenly while in office at New York City, on September 11, 1769. The duties of governor then fell on Lieutenant Governor Cadwallader Colden whose term was much less peaceful. Moore left with the respect of almost all the colony's leadership, the only exception were certain religious fundamentalists angered by his efforts to create a theater or playhouse.
Henry married Catherine Maria Long of another prominent Jamaica family in 1765. The couple had several children, and after his death, she moved to England. Catherines Peak (altitude 1158 meters) in Jamaica is named after her, as local legend reports her to be the first women to climb the peak.