Born in Copenhagen,Denmark as the son of Ditlev Vilhelm Klein (1793-1868) and Marie Kirstine Skousboe (1806-1891). He first trained as a stonemason before studying at the Royal Danish Academy of Fine Arts where he was awarded the grand silver medal in 1856. From 1851 to 1856, he worked as a draftsman for Michael Gottlieb Bindesbøll who he always considered to have been his main instructor.
From 1857 to 1862, he worked as an architectural designer for Ferdinand Meldahl who also influenced his style. He also worked for Johan Daniel Herholdt during the construction of Selchausdal Manor (1857) in Kalundborg, thus receiving insights for the most prominent architects of the day.
Klein was particularly fond of the Rosenborg style based on the Dutch Renaissance trends under Christian IV, crediting Bindesbøll for its prominent place in Danish architecture. From 1866 to 1872, he undertook substantial extensions to Bindesbøll's Lægeforeningens boliger social housing development in the Østerbro District of Copenhagen. His masterpieces include the Industriforeningen Building in the Italian Renaissance style on Vesterbro (erected 1872 in conjunction with the Nordic Exhibition, demolished 1976-77), the Danish Museum of Art & Design (Kunstindustrimuseet) in the Rosenborg style on Rådhuspladsen (1893), and the main building at the Ny Carlsberg Brewhouse (1901) also in Italian Renaissance style.
- "Vilhelm Klein", Den Store Danske. (in Danish) Retrieved 15 January 2013.
- "Vilhelm Klein". Kunstindeks Danmark & Weilbachs Kunstnerleksikon. Retrieved January 1, 2019.
- "Vilhelm Klein", Dansk Biografisk Leksikon. (in Danish) Retrieved 15 January 2013.
- "Selchausdal Manor House". visitvestsjaelland.dk. Retrieved January 1, 2019.
- "Lægeforeningens Boliger". dac.dk. Retrieved January 1, 2019.
- "1872 Nordic Industry and Art Exhibition". jdpecon.com. Retrieved January 1, 2019.
- "Brumleby" Archived December 11, 2012, at the Wayback Machine, Kulturkanonen. (in Danish) Retrieved 14 January 2013.