Robertson was born in Auchterarder, Scotland, and educated at Glasgow University where he obtained his PhD in 1926. From 1928 until 1930 he studied at the University of Michigan and then worked at the Royal Institution throughout the 1930s. After brief periods at the University of Sheffield and with Bomber Command of the Royal Air Force, he returned to Glasgow in 1942 and served as professor of chemistry until his retirement in 1970.

Robertson was one of the key figures who, centered on the Braggs and the Royal Institution, developed x-ray crystallography in the interwar period into one of the basic tools of both the physical and life sciences. He established structures for a large number of molecules, including accurate measurements of bond length in naphthalene, anthracene, and similar hydrocarbons. He also worked on the structure of the important pigment phthalocyanine (1935), durene (1933), pyrene (1941), and copper salts (1951). A notable contribution to the technique was his development of the heavy-atom substitution method, which he used in his investigation of phthalocyanine. This involves substituting a heavy atom into the molecule investigated. The change in intensity of diffracted radiation gives essential information on the phases of scattered waves.

In 1953 Robertson published a full account of his work in his Organic Crystals and Molecules in which he demonstrated the growing success in applying the new techniques of x-ray crystallography to complex organic molecules.