Albertus Welcker was born on 11 January 1884 in Alkmaar as a son of Ir. Johan Wilhelm Welcker, chief inspector general at the Department of Waterways and Public Works, and Clara Albertina de Wit. Albertus Welcker studied medicine in Leiden and after his finals in 1911 he became an assistent physician at the Wilhelmina Gasthuis in Amsterdam, where he specialized as a surgeon. His registrarship was interrupted for one year in 1912, when he volunteered for Dutch Red Cross field hospitals in Bulgaria and Turkey during the First Balkan War. In 1915 he established himself as a surgeon and in 1916 he obtained his doctorate (cum laude) with a dissertation on gangrene, based on his experiences in Bulgaria.

In 1919 he married Johanna Maria van Wessem. From this marriage, four children were born. During the Second World War, when the University of Amsterdam was closed, he clandestinely lectured surgery. In 1948 he retired as a hospital surgeon and in 1953 he gave up his private practice.

Despite his demanding position, Welcker still found time to devote himself to his studies of Dutch art. He started his art-historical career as a collector of contemporary prints during the First World War but sold this collection again at auctions in 1944 and 1948. In 1927 he bought his first old master drawings. Between 1932 and 1955 he published 31 articles on art, often in relation to drawings in his own collection.

In September 1955 Welcker was diagnosed with a terminal illness. On the occasion of the Rembrandt Year and at the suggestion of art historian Prof. Henri van de Waal, Welcker was awarded an honorary doctorate in arts and philosophy on 27 April 1956. This honour was bestowed upon him not only because of his studies on Dutch drawings, but also for the way - diligently and with a keen nose - in which he had built his important collection of Dutch drawings. On 11 July 1957, Welcker passed away.