He is known for his participation in the 1768–1771 first voyage of James Cook aboard HMS Endeavour, where he was one of the artists in the entourage of botanist Joseph Banks.

All of his drawings from the voyage were taken by his employer Joseph Banks and are now in the British Library and the Natural History Museum, London.

According to the art historian Bernard Smith, Buchan was not academically trained in figure drawing, but had likely learned through self-study by making copies of engravings.

[4] On the first voyage of James Cook, the naturalist Joseph Banks was accompanied by a party of eight people,[9] including the two artists Sydney Parkinson and Alexander Buchan, with the latter engaged for landscape and figure drawing.

[15] In Banks' journals, his first appearance is a note that Buchan drew views of the Cape Verde islands on 30 September 1768.

On 20 January, Banks visited an Ona village, very likely together with Buchan, whose drawing of An Indian Town at Terra del Fuego (engraved as View of a Village in the Bay of Good Success, in the Island of Terra del Fuego) illustrates Banks' description:[20] The town itself was situate upon a dry Knowl among the trees, which were not at all cleard away, it consisted of not more than twelve or fourteen huts or wigwams of the most unartificial construction imaginable, indeed no thing bearing the name of a hut could possibly be built with less trouble.

[21] From Tierra del Fuego, the Endeavour continued to Tahiti in order to observe the 1769 transit of Venus, arriving at the island on 13 April.

Banks wrote of his employee's death: I sincerely regret him as an ingenious and good young man, but his Loss to me is irretrevable, my airy dreams of entertaining my freinds in England with the scenes that I am to see here are vanishd.

[32][33] Buchan's best-known works are his illustrations of the people of western Tierra del Fuego, made during a stopover at the Bay of Success.

Cipriani added additional figures to Buchan's Inhabitants of the island of Tierra del Fuego, in their hut before the images were engraved by Francesco Bartolozzi.