Alexander Cullen was born on 18 June 1856 at Craigneuk, Wishaw, the son of William Cullen and his wife Mary Hill. Initially he was apprenticed to a builder, but he then attended Glasgow University and set up practice in Clyde Street, Motherwell in the early 1880s.

In 1889 Colonel H H Robertson Aikman commissioned him to reconstruct The Ross at Hamilton as a major house in the David Bryce manner: it was exhibited at the RSA and the RGI and was widely published, establishing his reputation. An office in Quarry Street, Hamilton was established to build it and undertake other work in the Hamilton area, Cullen having become architect to the standing joint committee of the County of Lanarkshire.

Cullen was admitted FRIBA in November 1898 on the recommendation of the Glasgow Institute of Architects: his proposers are not known as the relevant sheet is missing. By that date he had already been admitted FRSE, a very unusual distinction for a provincial architect.

In 1898-99 Cullen designed Brandon Chambers, Hamilton as his main office and Motherwell became a branch office. His practice prospered and by 24 June 1902 the business was so extensive with many commissions both public and private in the counties of Lanarkshire and Renfrewshire that he took into partnership James Lochhead and William Brown. Lochhead had been born in 1870 and articled to Allan Stevenson of Ayr, subsequently working for Henry Higgins of Glasgow and for the Office of Works before joining Cullen as a chief draughtsman and prospective partner. After Lochhead's arrival the work of the practice became extremely accomplished Edwardian Baroque and classical. Brown was the practice's long-time senior assistant. Born in 1873, he had served his articles with Cullen, remaining as an assistant before being taken into partnership. He was put in charge of the Motherwell office at 4 Clyde Street. Initially the practice name remained as Alex Cullen; it won the major competition for Hamilton Municipal Buildings in 1903, and a Glasgow office was established at Atlantic Chambers in Hope Street to supervise the Western District Hospital, a by-product of the competition for the Eastern District Hospital in which Cullen and Lochhead had been placed first but did not receive the commission. From 1908 onwards the practice was known as Alex Cullen, Lochhead & Brown.

Cullen had wide interests and was well read in architectural history as well as contemporary architecture and this is reflected in the varied nature of the practice's large library. In 1907 he gifted a proportion of this library to Hamilton District Libraries. Although these books have now been dispersed, the catalogue shows that in addition to the publications of Pugin, Scott, Fergusson, the Audsleys, Starforth and G A T Middleton, Cullen subscribed to 'The American Architect' and bought books on Japanese and Mexican architecture; and among his continental books was a copy of Guimard's 'Le Castel Beranger'. Cullen himself was the author of many articles on a variety of subjects and his book 'Adventures in Socialism', a history of New Lanark, was published in 1910. He was a Fellow of the Society of Antiquaries of Scotland and at the time of his death he was President of the Architectural Section of the Glasgow Philosophical Society.

Cullen died at the relatively early age of 54 on 21 January 1911 at 3 Blythswood Square, and was buried at Bent Cemetery, Hamilton. By that date a Glasgow office had been established in Atlantic Chambers in Hope Street and he had a house at 3 Blythswood Square. His moveable estate amounted to £9,238 11s 2d, a very large sum for an architect at that date. He was survived by his widow, Barbara Rodger, and by three sons and two daughters. His youngest son, Alexander (born 1892), was in the first year of his apprenticeship with the firm at the time of his father's death. Anither son, James, appears to have also been an architect. Lochhead and Brown continued the practice, which was known thereafter simply as Cullen, Lochhead & Brown.