Williams, who was born in Santa Rosa, California, received a PhD in physics from Cornell University in 1935 and then went on to teach astronomy at the University of Michigan, transferring to the physics department in 1945. He remained there until 1950, when he transferred to a lecturing post in the biochemistry department at the University of California, Berkeley. In 1964, when a department of molecular biology was established at Berkeley, Williams became its chairman.

As an astronomer Williams worked on the estimation of stellar surface temperatures. Military research during World War II turned his attention to electron microscopy, and an insight drawn from his knowledge of astronomical techniques led to a fruitful collaboration with the crystallographer Ralph Wyckoff. The early electron microscopes were transmission microscopes, i.e., the beam of electrons passes through the sample, giving a two-dimensional image. Working with Ralph Wyckoff at Michigan, Williams developed a technique of preparing specimens so that they could be observed with reflected beams of electrons. The technique involves depositing metal obliquely on the specimen. This effectively ‘casts shadows’ and creates a vivid three-dimensional effect in the image.

Williams turned to the study of viruses, using his shadowing technique, and made important contributions to an understanding of viral structure. In 1955 (in collaboration with Heinz Fraenkel-Conrat) he achieved, with the tobacco mosaic virus, the first reconstitution of a biologically active virus from its constituent proteins and nucleic acids.