Joachim Brehm was apprenticed from 1803 to 1807 to Ernst F. Rumpf, Professor of Chemistry and Pharmacy at the Royal Bavarian College at Durckheim, who was also a practising pharmacist. He then served as assistant pharmacist with different firms in Durckheim, Rastatt, Geneva, Strasbourg, and Bern. In 1816 he came to the Cape of Good Hope via England and worked for Dr. F.L. Liesching* until 1820. Lieshing practised as a physician, but also had an apothecary shop, both in Loop Street, Cape Town. In 1820 Brehm was registered by the Supreme Medical Committee as an apothecary, chemist and druggist. He moved to Uitenhage in the same year. There he established an apothecary shop and also provided medical services. Hence he was known locally as "Dr Brehm", though he was not qualified or registered as a medical practitioner. He married Anna Maria Menne, who died in December 1843 at the age of 53. Later he became deputy sherrif of Uitenhage and carried on an export business in hides, skins, and horns, as well as a property agency. His son, George B.A. Brehm (born 1823) also became an apothecary.
Brehm was interested in botany even before he came to the Cape. He collected plants in the vicinity of Cape Town, experimented with their medicinal properties and sent collections of plants, seeds and bulbs to Prof F. de P. Schrank of Munich, who published papers on them in 1822 and 1824. In Uitenhage he established a garden that was long renowned in the eastern districts and drew many distinguished visitors., One of these was Dr. Nathaniel Wallich*, a visiting botanist from India in 1843, who praised Brehm's garden and his botanical and general knowledge. The botanical collector Carl L. P. Zeyher* worked for Brehm as a gardener for some time. In January 1856 the "Hollandsche Maatschappij van Landbouw" awarded him a gold medal for having sent over to Holland a collection of plants of the genus Encephalartos. According to the horticulturalist William Tuck*, Brehm first imported the Bahia Naval Orange from Brazil in about 1850, more than 20 years before it was introduced into the United States (Webber, 1925, p. 15).
Brehm became a corresponding member of the South African Institution in Cape Town (the first truly scientific society in southern Africa) in January 1830. He corresponded with, and sent plants to, botanist William H. Harvey*, who named the genus Brehmia in his honour. He is also commemorated in the species Chlorophytum brehmeanum.