Kellett joined the Royal Navy in 1822.  He spent three years in the West Indies and then served on survey vessels under William Fitzwilliam Owen in Africa, as second in command of HMS Sulphur under Edward Belcher in the East Indies ,  and as captain of HMS Starling in the First Opium War with China during which he was promoted to Commander in 1841 and Post-Captain in 1842. 
In 1845 he was appointed captain of the survey ship HMS Herald as part of a hydrography survey mission whose primary mission was to survey the coast of North America from Guayaquil to Vancouver , including the Galapagos islands . He was temporarily reassigned in 1848 to join the search for Sir John Franklin . During this voyage he sailed through the Bering Strait across the Chukchi Sea and discovered Herald Island . Kellett landed on Herald Island and named it after his ship. He also sighted Wrangel Island in the western horizon. William Pullen was on this expedition.In 1852, he commanded HMS Resolute and went to the aid of Robert McClure , whose vessel, Investigator , was trapped in the Arctic.  His men constructed a storehouse on Dealy Island off the south coast of Melville Island . 
Several places in Hong Kong have been named after him: Kellett Island , Kellett Bay and Mount Kellett. Kellett Bluff on Henry Island , Washington, USA, was probably named after Captain Kellett as well. It is a place with extreme currents, views, and is frequented by feeding orcas. Kellet's whelk Kelletia kelletii is named after him.