The son of a Presbyterian minister in the Scottish capital Edinburgh, Crum Brown studied arts and then medicine and chemistry there, gaining an MA degree in 1858 and his MD in 1861. He was also awarded a doctorate from London University (1862) and worked with Robert Bunsen at Heidelberg and Hermann Kolbe at Marburg before returning to Edinburgh as a lecturer in 1863, becoming professor of chemistry in 1869.

Crum Brown was essentially a theoretician of organic chemistry and his structural formulae, introduced in his MD thesis On the Theory of Chemical Combination (1861) and taken up by Edward Frankland in 1866, are essentially the symbols used today. In 1867–68, with T. R. Fraser, he carried out pioneering work in what is now called structure/activity relationships in pharmacology. In 1892 (with J. Gibson) he proposed a rule (Crum Brown's rule) concerning the effect of substitution of an organic group into a benzene ring that already contains a group. The rule can be used to predict the position into which the existing group will direct the second group. Other research interests were physiology (the function of the semicircular canals in the ear), phonetics, mathematics, and crystallography.