Købke was born in Copenhagen, where he trained at the Royal Academy of Fine Arts from 1822 to 32. From 1825-8 he was a pupil of the portrait painter C.A. Lorentzen, then studied under C.W. Eckersberg until about 1834.
In 1838 he travelled to Italy via Germany where he visited the painter J.C. Dahl in Dresden. He stayed in Rome, Pompeii and Naples, but spent much of his time sketching outdoors on the island of Capri.
Købke painted mainly landscapes and portraits. His landscape paintings are usually based on plein air studies and often depict subjects close to his home in Copenhagen. The majority of his portraits are of relatives or close friends.