Born in March 1735 in Crieff, Perthshire, he studied at Crieff Grammar School and then University of Edinburgh, and obtained a medical degree at St. Andrews (MD 1763). He served as an apprentice with G Dennistoun in Falkirk (1752-6), and became a navy surgeon in 1760.
In 1764 Wright became the assistant to a Dr. Gray in Kingston, Jamaica. He stayed on the island until 1777. He enrolled in the British Navy in 1779 and was captured by the French. He returned to Jamaica in 1782 and the following year became Physician in Chief of the colony. He returned to Edinburgh in 1785. He joined an expedition led by Sir Ralph Abercromby (1734–1801) exploring the Caribbean from 1796 to 1798.
William Wright became a Fellow of the Royal Society in 1778. He was a member in numerous societies, among them the Linnean Society of London of which he became associate member in 1807; the Wernerian Natural History Society in 1808, of which he was a founding member; the Royal College of Physicians of Edinburgh, over which he presided in 1801.
Wright published numerous articles in medicine. His Jamaican collections became an important contribution to natural history. Notably, he described more than 750 plant species.
In 1795 he was visited by Johann Gottfried Schmeisser.
He is buried in the north west section of the western extension to Greyfriars Kirkyard.
Greyfriars Kirkyard is the graveyard surrounding Greyfriars Kirk in Edinburgh, Scotland. It is located at the southern edge of the Old Town, adjacent to George Heriot's School. Burials have been taking place since the late 16th century, and a number of notable Edinburgh residents are interred at Greyfriars. The Kirkyard is operated by City of Edinburgh Council in liaison with a charitable trust, which is linked to but separate from the church. The Kirkyard and its monuments are protected as a category A listed building.