Richard Anthony Markham was born in Leeds, England, the son of Richard Markham, a cloth merchant. He attended the University of Edinburgh—possibly instructed by John Hope—and became friendly with James Edward Smith. He changed his last name to Salisbury following a supposed financial arrangement for support in his studies. This arrangement made with a Mrs. Anna Salisbury, related by marriage to his grandmother, or so he claimed in correspondence with Joseph Banks.
Salisbury married Caroline Staniforth in 1796. One child, Eleanor, was born to the couple in 1797; the two separated shortly thereafter. Salisbury had apparently misrepresented his finances when he had proposed marriage, and had large debts at the time of his daughter's birth and had declared bankruptcy for dubious purposes. His honesty in legal and financial matters seems to have been questionable, if not devious. He apparently recovered financially by 1802, when he bought a house.
He established substantial gardens at one of his father's estates, Chapel Allerton, near Leeds, and purchased the former estate of Peter Collinson, Ridgeway House. It was at the latter that a long running dispute began between Smith and him.
Salisbury contributed annotations to Edward Rudge's Plantarum Guianæ Icones (1805–7), and descriptions to Paradisus Londinensis (1806–9). The latter was illustrated by William Hooker, and contained the genus name Hookera honouring him. Smith improperly renamed the genus Brodiaea a few years later, after his wealthy "friend and patron", James Brodie of Brodie.
In 1809, Salisbury was appointed the first honorary secretary of the Horticultural Society. His successor Joseph Sabine found he had left the accounts in disarray. He moved to London around this time; his small garden contained a large number of exotic and rare plants.