Barrington was the fourth son of the first Viscount Barrington. He was educated for the profession of the law, and after filling various posts, was appointed a Welsh judge in 1757 and afterwards second justice of Chester. Though an indifferent judge, his Observations on the Statutes, chiefly the more ancient, from Magna Charta to 21st James I. (1766), had a high reputation among historians and constitutional antiquaries. In 1773 he published an edition of Orosius, with Alfred's Saxon version, and an English translation with original notes. His Tracts on the Probability of reaching the North Pole (1775) were written in consequence of the northern voyage of discovery undertaken by Captain C. J. Phipps, afterwards Lord Mulgrave (1744-1792).
Barrington's other writings are chiefly to be found in the publications of the Royal and Antiquarian Societies, of both of which he was long a member, and of the latter vice-president. Many of these were collected by him in a quarto volume entitled Miscellanies on various Subjects (1781). He contributed to the Philosophical Transactions for 1780, an account of Mozart's visit at eight years of age to London. In his Miscellanies on varied subjects he included this with accounts of four other prodigies, namely, William Crotch, Charles and Samuel Wesley, and Garrett Wellesley, Lord Mornington. Among the most curious and ingenious of his papers are his Experiments and Observations on the Singing of Birds, and his Essay on the Language of Birds. He was buried in the Temple Church, London.