The HMS Gaspee was apparently built as a single-masted sloop-of-war, part of a class of seven light, fast revenue cutters contracted by Admiral Colville for the British Royal Navy, and built in American shipyards (possibly some in Canada) along the Northeast coast.  The usual crew compliment was 19-26 sailors. According to the excellent Navy List database information kindly provided by CH Donnithorne, the Gaspee was purchased by an order of Lord Admiral Colville dated 07Jan1764.  For a more complete discussion on this topic, see: Rigging.html 

These revenue ships were devised to enforce the various trade laws (i.e., the Townsend Acts) of the British Empire as applied to American shipping.  As pointed out by Neil R. Stout in The Royal Navy in America, 1760-1775: A Study of Enforcement of British Colonial Policy in the Era of the American Revolution (Naval Institute Press, Annapolis, MD  1973, p59:

These little vessels, the original coast guard cutters, proved to be the most useful craft in the North American command, because their fore-and-aft rigging enabled them to sail in weather too severe for the full-rigged men-of-war, and their shallow drafts allowed them to go into coves and creeks after smugglers.

These ships were not likely to ever have been conceived as being able to reliably transit the open ocean or to actually engage in a naval battle, as they were too lightly armed. In fact, the whole class of ships was withdrawn, sailed off to England, and sold off as surplus once the burning of the Gaspee in 1772 made it evident to the Admiralty that war with the American colonies was a distinct possibility (see Sultana.html). Rather, their purpose was to intercept unarmed civilian ships suspected of smuggling contraband into and out of American bases. Overall command of these ships rested with the Royal Navy in the area of the Halifax, Nova Scotia and Boston, Massachusetts ports. In 1772, this command was under the direction of Admiral John Montagu.  

The American Colonist did not appreciate the interference this fleet had on their sea trade, and referred to them disparagingly as "Spanish guarda costas". But, as Stout in The Royal Navy in America, p138 explains:

The Captains' logs of the Gaspee are to be found in Royal Navy archives Admiralty Papers, Class 51: Accountant General's Department. Vol. 3853, 3856 and 4197.  From Admiralty records purchased (at great expense, might we add) from the National Archives (UK) we have what appear to be the re-copied pay records of the Gaspee, Ref#: ADM 33/645 and we get the following snippet:

Sandy Hook, NJ is located on the south coast entrance to New York harbor.  Cape May, NJ is located at the very southern tip of New Jersey, at the entrance of the Delaware River.  New Castle, Delaware is south of Wilmington, both on the Delaware River leading up to Philadelphia. We think that the now-defunct Rudy Island was in the Delaware River just west and south of New Castle, probably now filled in with development.  Although there are no references to this island obtained in a Google search, it is clearly marked on Fisher's  1776 Map of the Delaware River available on-line through the NY Public Library Digital Images Collection. Located at a narrow bend in the Delaware River, Rudy Island appears to be perfectly located to spot and interdict merchant ship traffic coming from both directions.

Left. The Hartford Courant, Monday December 24, 1764. Pages 2 & 3 

Over her 106 month existence, over 230 men had been variously assigned to the Gaspee at one point or another. Of interest, of her 230 crewmen, over 142 deserted the ship, and 8 died at one point or another; that's a 62% desertion rate, and a 4% death rate, and an attestation to the poor living and working conditions on the ship. The motivations for desertion were many.  Some were impressed seamen, taken from dockyards areas or from other merchant ships and forced involuntarily to work on His Majesty's Ships.  Such men were unlikely to stay when opportunity to escape presented itself.  Others may have been motivated by a ticket to travel elsewhere, perhaps hoping to escape from Europe to new opportunities available in the frontier of the New World.  Note that most crewmen were in their early twenties, full of hope for a new and exciting life, and full of hormones which could lead them astray at the very next port. Some may have been starving and looking for room and board however temporarily. As even in more modern times, some may have been running from things in their lives, such as wives, ex-lovers, debts, and the law.  Aliases could be easily taken, and most men were taken at their word.  After jumping ship, they need not fear recapture; the land was too vast for most search parties.  Impressment crews would rather take the first person they found that could serve as an able-bodied seaman, rather than search endlessly for someone who had deserted. Some of those who deserted undoubtedly ended up on other ships (probably at much higher pay), perhaps under different assumed names, while others may have stayed in America and assimilated into our society.  For an excellent review of life aboard a Royal Navy schooner, the HMS Sultana--a simaller sister ship of the Gaspee, see Porter's "Wood, Water, and Beef".

We know that the Gaspee plied all of the waters along the Northeast American coast long before she met her untimely demise in Narragansett Bay. After her post-commissioning sea trials in Halifax, she was next stationed out of Philadelphia, PA in July and August 1764, then took up station near Portland, ME (Casco Bay) until the spring thaw in 1765. The earliest outside reference we have found of the existence of the HMS Gaspee relates to an adventure in Casco Bay in December 1764: From  Stout, p62-63:

Lieutenant Thomas Allen, her commander, found vessels arriving and leaving "without so much as taking the least notice of the Custom house." Allen seized two or three ships, but the Customs Collector of Falmouth refused to hold them for prosecution. Meanwhile, several of the Gaspee's crewmen deserted, and one of her seizures, taking advantage of the depletion, made her escape. Left without enough hands to navigate the Gaspee safely, Allen impressed four merchant seamen, but a mob kidnapped him and made him release his new recruits. The lieutenant finally decided that "it would be no manner of purpose for him to remain longer there," so he made a last trip to the Falmouth customs house to turn his remaining two captures over to the collector. One of his oarsmen deserted while Allen was completing the transaction. The Gaspee limped back to Halifax minus seven of her best seamen, one midshipman, and her boatswain's mate, and, of course, with no prizes.

In summer of 1765, the Gaspee was dispatched from Halifax to Boston to render aid during the anti-taxation riots that eventually resulted in the abdication of Massachusetts Colonial Governor Francis Bernard. The Gaspee then intercepted a ship carrying the hated tax stamps and conveyed the cargo to safety at the British fort at Castle William out in Boston harbor (Stout, p92-93). 

Another early record of the existence of the HMS Gaspee comes from the last will and testament of one of its crew, apparently based in the New York City. From:  Tami, Chris. New York City Wills, 1706-90 Vol II. Ancestry. Inc. Orem, UT, 1999.

In the name of God, Amen. I, JOHN WALCOT, mariner, belonging to His Majesty's armed Sloop Gaspee, Thomas Allen, Commander, do for divers good causes, make this my last will and testament. I leave to my wife, Mary Walcot, of New York, all wages and money due to me from the Honorable Commissioner of the Navy, and whatever else I have, and I make her executor.

Dated October 28, 1765. Witnesses, Vincent Montanye, Samuel Dodge, Philip Pelton. (No Probate.)

The general irritation was aggravated by the supercilious behavior of the king's representatives and officers, military, naval, and civil.  They had always expressed contempt for the provincials as an inferior order of people; now they looked upon them as already rebels.  Numerous contemporary accounts may be found, in Graydon and other journal-keepers of the day, of the extremes to which this sort of thing was carried.  The captain of the royal armed schooner "Gaspee" (the same burned a year or two later by the New Englanders) was noted for his brutality. He and his officers maltreated Davis Bevan, a citizen of Chester County, put him in irons, and otherwise abused him. Bevan, in return, sued them for ill treatment.  The people were no longer in a frame of mind to submit to things which they would not have noticed five or six years before.  They quarreled with the captain of a sloop-of-war for firing a salute on arriving in port,  They accused the customs collector and naval officer of extorting illegal fees.  The women took part in the quarrel also, and it was every day more and more noticeable that the spirit of union was diffusing itself among the colonies, each part and section espousing as its own the grievance of every other part.

Shortly afterward (October, 1769) there was an affray on the Delaware River caused by the brutality of Capt. William Diddington, commander of the royal armed schooner "Gaspee," who, with David Hay, sized Davis Bevan, a citizen of Chester County, who had been fishing in the Delaware, near Chester, and maltreated him. This event added to the discontent.

The King’s ships are very vigilant on the American coasts. About the beginning of June, they spied a shallop belonging to Chester, laden with flour and lumber, and maltreated the master, who complained to the civil magistrate against the officer, and he was taken into custody of the sheriff, but rescued by the Captain of the King’s ship, who took the writ from the sheriff, and carried the man off in triumph.