Dilling was born in Aberdeen on 15 May 1886, the son of William (1846-1917) and Annie Dilling (1849-1931). He attended the local Ashley Road School and Robert Gordon’s College, and served as a volunteer in the Royal Army Medical Corps 1903-05. He graduated from the University of Aberdeen in 1907, M.B., Ch. B. In May of that year he became second assistant to Prof. J. A. MacWilliam in the Department of Physiology, before being appointed Carnegie research fellow in physiology. In 1909, as Carnegie research scholar in pharmacology, he travelled to Germany and served as first assistant to Rudolf Kobert, professor of pharmacology and physiological chemistry at the University of Rostock.

Returning to Scotland in 1910, he was appointed lecturer in pharmacology at the University of Aberdeen. Here he delivered a course on experimental pharmacology for medical students - the first of its kind in Britain. In 1914 he was appointed to the new Robert Pollok lectureship in Materia Medica and Pharmacology at the University of Glasgow, though this was interrupted by war service; during the First World War he returned to the Royal Army Medical Corps (R.A.M.C).

In 1920 he became a lecturer in pharmacology at the University of Liverpool, rising to Associate Professor in 1926 before his appointment to the newly created Chair of Pharmacology in 1930. He served as Dean of the Faculty of Medicine during 1924-35 and 1939-45. During the Second World War he commanded the medical company of the University Senior Training Corps, working alongside student stretcher bearers to receive casualties at the railway terminus.

Outside the university, he was nominated to the General Medical Council in 1938, becoming Chairman of its Pharmacopoeia Committee in 1948, and was a Privy Council representative on the Council of the Pharmaceutical Society (1938). A keen music lover, he was Chairman of the Liverpool Philharmonic Society from 1936 and chairman of the local branch of the British Music Society from 1931. He was also chairman of the Young People’s Opera Circle and gave numerous talks to students of opera.

Dilling lectured and wrote extensively on various physiological and pharmacological subjects, including medical history. He also edited Dr Mitchell Bruce's 'Textbook of Materia Medica'; under the revised title of 'The Pharmacology and Therapeutics of the Materia Medica' this book had reached its 18th edition by 1944, and he had almost completed the 19th edition at the time of his death. From 1940 on he contributed the chapter on recent developments in drug therapy to the annual 'Medical Progress' volume of the 'British Encyclopaedia of Medical Practice'. With Samuel Hallam he was joint author of another textbook, 'Dental Materia, Medica, Pharmacology and Therapeutics'.

He married Vida Ducat in August 1914 and they had two daughters: Nancy (1923-1975) and Eva (1926-2000). He died in Coniston, 18 August 1950.