Born in Dundee, Henderson began work as an attorney's clerk who made a reputation as an amateur astronomer. In 1831 he accepted an appointment as director of a new observatory at the Cape of Good Hope in South Africa. While observing Alpha Centauri he found that it had a considerable proper motion. He realized that this probably meant that the star was comparatively close and a good candidate for the measurement of parallax – the apparent change in position of a (celestial) body when viewed from spatially separate points, or from one point on a moving Earth. All major observational astronomers had tried to detect this small angular measurement and failed. Henderson at last succeeded in 1832 and found that Alpha Centauri had a parallax of just less than one second of arc. The crucial importance of this was that once parallax was known, the distance of the stars could be measured successfully for the first time. Alpha Centauri turned out to be over four light years away. Unfortunately (for Henderson), he delayed publication of his result until it had been thoroughly checked and rechecked. By this time Friedrich Bessel had already observed and published, in 1839, the parallax of 61 Cygni.

In 1834 Henderson became the first Astronomer Royal of Scotland.