Born on Merseyside at Birkenhead (13 January, 1859 GRO), the artist studied in Germany, in London at Heatherley's, and in Paris at Julian's Atelier, working primarily in watercolour. He came to Newlyn from Polperro, where he first painted in c1890, but may have been present earlier than this.

Stanhope FORBES remarked wittily that Rheam had been 'imported' to bulk up the Newlyn cricket side. By the 1891 Census he was living at St Peters, Newlyn (aged 32 years) as a boarder, with Samuel Green ENDERBY boarding in the same house. A first cousin of Henry Scott TUKE, Rheam was so pleased with Newlyn that he remained for the rest of his life.

A staunch Quaker, his paintings were in a romantic, late Pre-Raphaelite style. At the Opening Exhibition of NAG (1895) a reviewer commented, "Among the watercolour men who choose figure subjects Mr Rheam is conspicuous; his Belle dame sans merci, which was sold, is as complete a realisation of the heroine of Keat's poem as any artist is ever like to give us." He also showed Wrecked, At the Window and Gorse. In that same year at the 'Sketch Exhibition' he showed seven pieces of work and sold them all, the best seller of the show.

In 1897 he lived at Boase Castle Lodge, Belle Vue in Newlyn, which he and his wife Alice Elliott Rheam took over from the Madderns (who gave up their lodging house after many years). In that same year, Rheam became the Hon Secretary/Curator of the Newlyn Society of Artists (NSA) and continued loyally in that post until his death.

In 1903, amongst other exhibits, he sold both Sketch for Pandora and Melisande to the then Bishop of Ripon. His curatorship meant quite arduous administration duties in addition to an active painterly and cricketeering life. The Rheams remained in Newlyn until about 1914 when they moved to West Lodge in Alverton (Penzance). Henry continued to visit Lamorna until his death in 1920.