She was one of the vessels in the First Fleet, that the British government hired to transport convicts for the European colonisation of Australia[broken anchor] in 1788.

On her return voyage from Australia the British East India Company permitted her to carry a cargo from Canton back to Britain.

The cause of the fever was likely inadequate management of the bilge, as reported by John White, the surgeon aboard HMS Sirius in June 1787: The illness complained of was wholly occasioned by the bilge water which had by some means or other risen to so great a height that the panels of the cabin and the buttons on the clothes of the officers were turned nearly black by the noxious effluvia.

[12]Complaints by Surgeons White and Balmain to First Fleet captain Arthur Phillip led to regular pumping of Alexander's bilge thereafter, with a corresponding improvement in convict health.

A band of five convicts and a number of able seamen had armed themselves with iron bars, intending to overpower the guard and sail the vessel to the nearest landfall.

[19] After the decision was made to move the site of the colony, Alexander arrived at Port Jackson, Sydney, Australia, on 26 January 1788 to unload her convicts.

At Port Jackson Henry Kable, a convict, successfully sued Duncan Sinclair for the loss of his possessions during the voyage.

The British East India Company had hired Alexander in 1786 to carry tea from Canton after she had disembarked her convicts.

She left Port Jackson on 14 July 1788 in company with Friendship, whose crew she picked up when that ship was scuttled at Batavia on her way to Canton.

Alexander carried with her to England the last papers of the French navigator Lapérouse, whose expedition's two ships had most likely already shipwrecked on the fringing reef of the island of Vanikoro in the Solomon Islands, as the French expedition had departed Botany Bay in March 1788 to head North with the intention of returning to Europe via the Torres Strait.

Issues of Lloyd's Register for 1799 to 1801 described Alexander as built in 1783 in Hull, of 468 tons burthen, and as trading between London and Petersburg.