As to what they would have been doing... they were part of the British naval squadron maintained in the West Indies, basically to act as both a support to the colonial authorities, but also to protect British interests - there was a war going on in Venezuela, and much of the South American coast was regarded as lawless, and the U.S. civil war was also underway. British naval ships were also very active in suppressing slavery in the first half of the 19th century, and although most countries had officially outlawed slavery by this time, the R.N. patrolled to make sure that this was enforced
Murray and the Skipjack gets some mentions at this time in The Milne Papers vol.2 (a collection of letters etc. from Admiral Milne, then commanding the station). From this it is clear that she was based at Barbados. Milne states that she was there "For the protection of British interests and Commerce"
She had an interesting episode in in February 1860 when Murray was informed that 2 British subjects had been seized as hostages by 'insurgents', and immediately went to the reported location, in the river Orinoco; the Skipjack went 40 miles upriver and after confronting the 'insurgents' (which Murray described as 'very contemptible') they undertook to release the hostages . The ship unfortunately ran aground in the shoal river, and was with much effort by the crew that they got free
Admiral Milne later wrote: 'You have a very superior man in Murray.. very unassuming, modest and shy but well read and a sensible, good officer, devoted to the service and his gunboat is a perfect man of war..."