Background
His father and twin brother (John Cleveley the Elder, c1712–1777, and John Cleveley the Younger, 1747–1786) were also artists, with John the Younger (and possibly Robert too, to judge from his style) gaining some training in watercolours from Paul Sandby, previously a teacher at the Royal Military Academy, Woolwich.
Career
John the Elder had tried and failed to make a living in working in a dockyard, and so did Robert, as a caulker. However, mocked by other dockyard workers for wearing gloves whilst working, John did not enjoy his time there, giving it up and in 1770 volunteering for the navy as a clerk. The Asia returned in 1777, and from then to the end of his life Robert followed a double career as purser on board various ships stationed in the Home Fleet (though most probably exercising his functions through a deputy for some or all of the time) and as a marine painter.
He did, however, still make occasional voyages with Vandeput, such as when he served as ‘assistant to the clerk of the kitchen’ in the royal entourage when the royal yacht Princess Augusta (under Vandeput) took Prince William Henry, later Duke of Clarence, to Hanover in July and August 1783.
This moving in royal circles later brought him an appointment as Marine Draughtsman, first to the Duke of Clarence and then to the Prince Regent. He died in 1809 in Dover after accidentally falling down a cliff.