Born in Dundee, Scotland, he was trained as a lawyer and worked his way up through the profession as an assistant to a variety of nobles. However, his major hobbies were astronomy and mathematics, and after coming up with a new method for using lunar occultation to measure longitude he came to the attention of Thomas Young, superintendent of the British Navy's "Nautical Almanac". Young helped Henderson enter the larger world of astronomical science, and on his death a posthumous letter recommended to the Admiralty that Henderson take his place.
Henderson was passed over for that position, but the recommendation was enough to get him a position as the British observatory at the Cape of Good Hope in South Africa. There he made a considerable number of stellar observations between April 1832 and May 1833, including those for which he is remembered today. It was pointed out to him that the bright southern star Alpha Centauri had a large proper motion, and Henderson concluded that it might be a close star.
The 1830s version of the "space race" was to be the first person to measure the distance to a star using parallax, a task which is easier the closer the star. Henderson was thus in a good position to be this person. After retiring back to the United Kingdom due to bad health, he began analyzing his measurements and eventually came to the conclusion that Alpha Centauri was just slightly less that one parsec away, 3.25 light years. This figure is reasonably accurate, being 33.7% too small.
Doubts about the accuracy of his instruments kept him from publishing, however (there had been previous, discredited attempts to claim a measurement of stellar parallax), and eventually he was beaten to the punch by Friedrich Wilhelm Bessel, who published a parallax of 10.4 light years (9.6% too small) for 61 Cygni in 1838. Henderson published his results in 1839, but was relegated to second place because of his lack of confidence.
In the meantime, his measurement work at the Cape had led him to be appointed the first Astronomer Royal in 1834. The private observatory on Calton Hill in Dundee -- where he had made his first observations as an astronomer -- was sold to Edinburgh University and the vacant chair of astronomy there given to him on the advice of Prime Minister Lord Melbourne.