Wilkins was born at Pongaroa in New Zealand. After graduating in physics from Cambridge University in England in 1938, he joined John Randall at Birmingham University to work on the improvement of radar screens. He received his PhD in 1940 for an electron-trap theory of phosphorescence and soon after went to the University of California, Berkeley, as one of the British team assigned to the Manhattan project and development of the atomic bomb. The results and implications of this work caused him to turn away from nuclear physics and in 1945 he began a career in biophysics, firstly at St. Andrews University, Scotland, and from 1946 at the Biophysics Research Unit, King's College, London.

The same year that Wilkins joined King's College, scientists at the Rockefeller Institute announced that genes consist of deoxyribonucleic acid (DNA). Wilkins began studying DNA molecules by optical measurements and chanced to observe that the DNA fibers would be ideal material for x-ray diffraction studies. The diffraction patterns showed the DNA molecule to be very regular and have a double-helical structure. The contributions of Wilkins's colleague, Rosalind Franklin, were especially important in showing that the phosphate groups are located on the outside of the helix, so disproving Linus Pauling's theory of DNA structure.

Wilkins passed on his data to James Watson and Francis Crick in Cambridge who used it to help construct their famous molecular model of DNA. For their work in elucidating the structure of the hereditary material, Wilkins, Watson, and Crick were awarded the 1962 Nobel Prize for physiology or medicine.

Wilkins went on to apply his techniques to finding the structure of ribonucleic acid (RNA). From 1955 he was deputy director of the Biophysics Research Unit and from 1963 he was professor at King's College, firstly of molecular biology and from 1970 of biophysics. He retired in 1981.