Henry was born at Wrexham, and originally trained as an apothecary. He practised at Knutsford and Oxford, before moving to Manchester in 1764. He was in partnership in Manchester with James Ainsworth, a surgeon. In 1778, he was appointed visiting apothecary at MRI. He was a founder and lecturer in chemistry at the Manchester College of Arts and Sciences (instituted 1783) and a member of Manchester Literary and Philosophical Society. Henry is best known as a research chemist, inventing a new process for magnesium oxide and establishing a firm to produce magnesia on a commercial scale. He was also involved with sanitary reform in Manchester, and worked with Thomas Percival and John Ferriar on the Manchester Board of Health. Henry was an active member of Manchester Literary and philosophical Society, serving at its president from 1807-1816 He was elected to the Royal Society in 1775.
He was the father of William Henry and possibly also of Thomas Henry (born before 1770, died 1798, a member of Manchester Literary and Philosophical Society).