Rasmussen was born at Ærøskøbing in Ærø, Denmark. He was the son of Johan Aranth Rasmussen and Caroline Sophie Kaasden. He was the eldest of eleven children born to a master tailor who later became City Treasurer. At fifteen, he went to Copenhagen to learn the hosiery trade and completed his training in 1861. While there, however, he was attracted to the art works he saw and took drawing lessons with the architects Hans J. Holm (1835–1916) and Christian Vilhelm Nielsen (1833-1910). Later, he took lessons with the animal painter Didrik Frisch (1835–1867).   
In need of some immediate income, he took passage as a cabin boy on a relative's merchant ship and visited Scotland; painting several works along the way. Upon returning, he enrolled in courses at the Copenhagen Technical College to improve his understanding of perspective.
He studied until 1866 with the landscape painter, Carl Frederik Aagaard (1833–1895). His first showing was at the Charlottenborg Spring Exhibition in 1863. He would exhibit there regularly until his death.
Inspired by his sailing experience, he visited Greenland in 1870-71, making sketches that later became paintings, one of which was purchased for the Royal Collection. In 1872, he was able to obtain a scholarship to the Royal Danish Academy of Fine Arts, which led him through the Netherlands, Belgium and Italy and galleries in Paris, Dresden and Berlin.
By 1880, he had become successful enough to get married to his cousin Anna Egidia Rasmussen (1852-1931) with whom he would have eight children. He bought a home and establish a studio in Marstal on Ærø. His first work there was a large mural at Marstal Church, depicting Jesus calming the waters, which featured portraits of prominent local citizens.
In 1893, he was returning from Greenland on the brig Peru when he apparently fell overboard and drowned. He died between Orkney and Shetland. He was last seen by the helmsman, standing aft with his easel. When he was reported missing, an extensive search of the area was conducted, but they were unable to find him.
- Sigurd Schultz. Carl Rasmussen Dansk Biografisk Leksikon, Gyldendal.
- Hanne Poulsen J.E.C. Rasmussen Danmark & Weilbach Kunstnerleksikon.
- Elisabeth Fabritius. "Chr.V. Nielsen". Kunstindeks Danmark & Weilbach Kunstnerleksikon. Retrieved March 1, 2019.
- Anne Lise Thygesen. "Hans J. Holm". Den Store Danske, Gyldendal. Retrieved March 1, 2019.
- Peter Nørgaard Larsen. "Didrik Frisch". Kunstindeks Danmark & Weilbach Kunstnerleksikon. Retrieved March 1, 2019.
- Lilli Lehmann. "C.F. Aagaard". Kunstindeks Danmark & Weilbach Kunstnerleksikon. Retrieved March 1, 2019.
- "Marstal Church, Aeroe". visitaeroe.com. Retrieved March 1, 2019.
- Anders Østergaard, Jacob Hald Pedersen, May Schack. "Carsten Jensen". Den Store Danske, Gyldendal. Retrieved March 1, 2019.CS1 maint: Multiple names: authors list (link)
- "Marstal Søfartsmuseum, Aeroe". visitaeroe.com. Retrieved March 1, 2019.
- "Svendborg Museum". svendborgmuseum.dk. Retrieved March 1, 2019.
- Helle & Benno Blæsild (1993) Marinemaleren Jens Erik Carl Rasmussen (1841-1893) (Svendborg og Omegns Museum) ISBN 87-87769-58-1.
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