Since he was a captain in her majesty's navy there will be some records on his service at The National Archives
see their research guides
http://www.catalogue.nationalarchives.gov.uk/RdLeaflet.asp?sLeafletID=287I only say some because his death was in 1800 so the earlier records to do not give such good coverage. There also seems to be some records on the ship itself.
You could try an index such as for 'The Times' which is likely to have had a newspaper report on the sinking.
Or you could make it easy on yourself and do a Google search for him and find
http://pages.quicksilver.net.nz/jcr/~vancouver2.html#James%20Hansonwhich gives the following information
James Hanson
James Hanson was born in London. His older brother, John Hanson, was a lawyer, who was known as the solictor and business agent of George, Lord Byron, the poet. While the poet was still a boy, he would stay with the Hanson family in Earls Court, London and he was well-known to James Hanson.
Hanson was selected for the expedition to the Northwest Coast of America. On 17 November 1790, Hanson became a lieutenant and joined the Chatham on 28 December 1790 in that rank as second-in-command to William Broughton, the ship's commander. After the storeship Daedalus arrived at Nootka and Vancouver learned of the death of its commander, Richard Hergest, Hanson was appointed as the replacement agent (or commander) in his place on 29 August 1792. No log for Hanson has been located. Hanson Island in Queen Charlotte Sound is named for him.
Hanson was promoted to commander on 24 July 1795 and in 1800 he was in command of HMS Brazen. The Brazen had been a French privateer l'Invincible General Bounaparte that had been captured the previous April. Operating in the English Channel, Hanson took a prize off the Isle of Wight and sent it into Portsmouth. The following morning, 26 January 1800, Brazen was driven by a gale on to the Ave Rocks near Newhaven and was destroyed. Only one man survived and Hanson was among the dead. He left no will.
At some time after Vancouver's voyage Hanson had married. His widow, who was in an advanced state of pregnancy, offered a reward for the recovery of the captain's body. Hanson's arm was marked with an anchor picked out in gunpowder.
Whidbey called the southern point of Gray's Harbor on the Washington coast Point Hanson but the name was not retained and it is now known as Point Chehalis. However, an island deep in Queen Charlotte Sound, north of Vancouver Island, is still known as Hanson Island.
Regards
Valda