Educated at Marylebone grammar school and at Eton College, he proceeded to King's College, Cambridge, and was elected a fellow of this society in 1768. In 1771 he took holy orders, and afterwards visited many parts of Europe as tutor and travelling companion to various noblemen and gentlemen. In 1786 he was appointed vicar of Kingston-upon-Thames, and ~fl 1788 rector of Bemerton, Wiltshire. He also held the rectory of Stourton from 1801 to 1811 and that of Fovant from 1811 until his death. In 1791 he was made prebendary of Salisbury, and in 1804 archdeacon of Wiltshire. He married in 1803 Eleanora, daughter of William Shairp, consul-general for Russia, and widow of Thomas Yeldham of St Petersburg.
During a long residence at Bemerton Coxe was mainly occupied in literary work. His Memoirs of Sir Robert Walpole (London, 1798), Memoirs of Horatio, Lord Walpole (London, 1802), Memoirs of John, duke of Marlborough (London, 1818-1819), Private and Original Correspondence of Charles Talbot, duke of Skrewsbury (London, 1821), Memoirs of the Administrations of Henry Pelham (London, 1829), are very valuable for the history of the 18th century. His History of the House of Austria (London, 1807, new ed. 1853 and 1873), and Memoirs of the Bourbon Kings of Spain (London, 1813), give evidence of careful and painstaking work on the part of the author. The style, however, as in all his works, is remarkably dull.
His other works are mainly accounts of his travels:
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