Richard Basil Perkins was born in Morpeth, New South Wales, the only son of Richard Henry Perkins, a general practitioner, and his wife Helen (née Phillips). He attended St Aloysius College, where he was dux in 1929. He entered Sydney University Medical School, interrupting his course to obtain a BSc (Hons) in Physiology in 1933, before graduating MB BS(Hons II) in 1935. He then became a resident medical officer at Royal Prince Alfred Hospital, Sydney, for two years before proceeding to England in 1938. He became MRCP Lond in 1939 and was awarded the Chesterfield Medal at St John's Hospital for Diseases of the Skin in the same year. He thus became the first Australian dermatologist to be trained in the English tradition of first qualifying in internal medicine.

Upon the outbreak of war, Perkins returned to Australia and enlisted in the Australian Army Medical Corps, eventually becoming a Lieutenant Colonel commanding 21 Field Ambulance in 1942. He was, however, much happier in clinical work and relinquished his rank to become dermatologist to the 2/7 Australian General Hospital in Lae, where he served for two years. He wrote a booklet, Tropical Diseases of the Skin, for the guidance of medical officers. After the war he was appointed honorary assistant dermatologist to Royal Prince Alfred Hospital, eventually becoming head of the dermatology department. He was appointed visiting lecturer in tropical dermatology at the University of Sydney from 1951 to1956, later becoming lecturer in dermatology, from 1961 to 1965. In 1956 he was elected president of the Australasian College of Dermatologists.

Dick Perkins' activities were characterized by a quiet, meticulous, almost compulsive thoroughness. He was widely read and informed on some of the most erudite and unexpected subjects. His opinion was widely sought in the management of abstruse and difficult cases. In 1960 he was stricken with cancer of the lung, which was treated with surgery and radiotherapy, and which later necessitated the relinquishing of his academic activities. He continued in clinical practice until shortly before his death. In 1940 Dick married Kathleen Bull, who had been a theatre sister at Royal Prince Alfred Hospital and there were three sons and two daughters of this union.