Klug was born in Lithuania of South African parents, and moved to South Africa at the age of three. He studied medicine for a year at Witwatersrand University, Johannesburg, and then changed to study science. He moved to Cape Town in 1947 where he took a master's degree in crystallography. In 1949 Klug moved to England, where he worked at the Cavendish Laboratory, Cambridge. In 1954 he went to Birkbeck College, London, and was director of the Virus Structure Research Group there from 1958 to 1962, when he returned to Cambridge.

Klug originally developed a technique of improving results of electron microscopy by illuminating the micrograph with laser light. For micrographs of regular structures, a diffraction pattern is produced, from which extra information on the specimen can be obtained. At Birkbeck, Klug worked with J. D. Bernal on the polio virus. With Donald Caspar he went on to study small viruses. These are either rod-shaped or spherical, and consist of nucleic acids covered by a protein coat. Klug and Caspar developed a theory of how the coat could be formed by an arrangement of smallish quasiequivalent protein molecules.

At Cambridge, Klug worked on helical viruses showing how the protein units form. He went on to investigate the structure and action of transfer DNA in animal cells. More recently he has worked on chromatin in cells. Klug received the 1982 Nobel Prize for chemistry for his work in molecular biology.