See his autobiography (1956).
Armstrong was born and lived most of his life in Lewisham, a suburb of London. After finishing school in 1864 at age 16, he spent a winter in Gibralter, with a relative, for health reasons. In the spring of 1865, Armstrong returned to England and entered the Royal College of Chemistry in London, now the department of chemistry at Imperial College. Chemical training in those days was not lengthy, and at the age of 18 he was selected by Edward Frankland to assist in devising methods of determining organic impurities in sewage.
Armstrong pursued further studies under Hermann Kolbe at Leipzig, earning a Ph.D. in 1869 for work on "acids of sulfur." A permanent appointment in 1879 at City and Guilds of London Institute, now also a part of Imperial College, followed. At age 36, Armstrong became Professor of Chemistry at yet another Imperial College precursor, the Central Institution in 1884. It was here that he established a three-year diploma course in chemical engineering, "seeing the need for a more scientific attitude of mind among British industrialists
He already had started on the systematic synthesis, degradation, and structural constitution of many naphthalene derivatives in 1881, building on earlier work on benzene derivatives and Erlenmeyer’s proposal for the structure of naphthalene. W. P. Wynne was his most important collaborator; their 263 naphthalene samples, accrued over several decades, are now preserved at Imperial College as the Armstrong-Wynne Collection. This research on naphthalene gave much impetus to the synthetic dye industry.
In 1887, Armstrong became interested in classifying substituents of benzene in terms of their meta- and ortho-para directing influences. It was in a footnote to an article on that theme in 1890 that his centric formula for benzene first appeared. His six affinities acting within a cycle predated both the discovery of the electron and modern theories of aromaticity. Armstrong recognised that affinities have direction and are not merely point particles, and so he might be said to have anticipated parts of the wave mechanical theories of the 1920s.