Henry Priestley was born in Bradford, Yorkshire, on 19 June 1884, the son of Herbert Priestley, an accountant. The family emigrated to Australia in 1886 and settled in Sydney. After attending Newington College, Henry enrolled in the faculties of arts (1902), engineering (1903) and science (1904) at Sydney University. He majored in chemistry and graduated BSc in 1906 with second-class honours in chemistry III. In the same year he enrolled in medicine II and graduated (MB 1909, ChM 1910) with first-class honours and the University Medal.

While completing a one year internship at the Royal Prince Alfred Hospital in 1910, he won a Beit Memorial Fellowship that enabled him to work with Sir Charles Martin at the Lister Institute in London (1911-12) where he developed his lifelong interest in nutrition. Subsequently, he was appointed as an assistant at the newly opened Institute of Tropical Medicine in Townsville (1913-18). In 1915 he was awarded an MD with a medal and the Ethel Talbot Memorial Prize by the University of Sydney for a thesis entitled The Mechanism of the Agglutinin Reaction and in 1918 he was appointed by the University as lecturer and chief demonstrator in physiology.

In 1921 he became McCaughey Associate Professor of Physiology with special responsibility for biochemistry and in 1938 he became foundation professor of the subject, the same year in which he became a foundation Fellow of The Royal Australasian College of Physicians. He became head of a separate department of biochemistry in 1939, a post he retained until his retirement in 1948. He served as dean of the faculty of science from 1946 to 1948 and was a fellow of the senate in 1948. Priestley was well liked and respected by a generation of students who appreciated his dedication and conscientiousness during the period when Chapman and Davies were the professors of physiology. He was known as `Whispering Henry' because of his soft voice.

In 1912, Priestley married Katie Maitland Gray, daughter of the Reverend Canon Maitland Gray of Dudley, UK, and had two sons, John and Peter. He died on 28 February 1961, followed by his wife on the succeeding day. During his professional life Priestley held honorary appointments at the Royal North Shore and Royal Prince Alfred Hospitals. His most enduring monument at Sydney University is the BSc (Med) scheme which he fought to have established in 1948, one of his last acts before retirement.