Klug was born in Želva, Lithuania to Jewish parents. The family moved to South Africa when he was two. Klug graduated with a degree in science at the University of Witwatersrand and studied crystallography at the
University of Cape Town before moving to England, completing his doctorate at Trinity College, Cambridge in 1953.
He moved to
Birkbeck College, University of London, in late 1953, and started working with Rosalind Franklin in
John Bernal's lab. This experience aroused a lifelong interest in viruses. During his time there he worked on the structure of the tobacco mosaic virus.
In 1962 he moved to the newly built
MRC Laboratory of Molecular Biology in Cambridge. Klug used methods from X-ray diffraction to develop crystallographic electron microscopy. In this method, two-dimensional images of crystals taken from different angles are combined to make three-dimensional images of the target. He worked out the structure of important nucleic acid-protein complexes.
Between 1986 and 1996 he was Director of the Laboratory of Molecular Biology in Cambridge, and was knighted in 1988. He was elected President of the Royal Society, and served from 1995–2000. He was appointed to the Order of Merit in 1995.