Salt; the son of a physician; was born in Lichfield. He trained as a portrait painter; first in Lichfield and then in London under Joseph Farington and John Hoppner. In 1802 he was appointed secretary and draughtsman to George Annesley; Viscount Valentia. They started on an eastern tour; traveling to India via the Cape. Salt explored the Red Sea area; and in 1805 visited the Ethiopian highlands. He returned to England in 1806. Salt's paintings from the trip were used to the Lord Valentia's Voyages and Travels to India; published in 1809.

Salt returned to Ethiopia in 1809 on a government mission to explore trade and diplomatic links with the Tigrayan warlord Ras Wolde Selassie. On his return he published A voyage to Abyssinia; and travels into the interior of that country; executed under the orders of the British government in the years 1809 & 1810; and a collection of drawings entitled Twenty-four Views Taken in St Helena; The Cape; India; Ceylon; Abyssinia and Egypt.