His father, Hans Henrik Bøgh, was a school teacher.
In 1863, he married Cathrine Sophie Henriette Møller, the daughter of a District Bailiff (Herredsfoged [da]), the equivalent of a local judge.
After serving as a soldier in the First Schleswig War, he attended the Royal Danish Academy of Fine Arts, where he studied with Johan Ludwig Lund, and decided to specialize in animal painting.
From 1860 to 1861, he made a study trip abroad, with the travel scholarship from Academy; visiting Brussels and Antwerp, but spending most of his time in Paris.
In 1870 and 1875, some of his works were purchased by the "Royal Painting Collection" (now the National Gallery of Denmark).