Osthaus was born in Hildesheim, Hannover (now Germany), on 5 August 1858. He studied at the Royal Arts Academy in Düsseldorf, Germany, with Andreas Muller, Peter Jansen, E. von Gebhardt, Ernst Deger, and Christian Kroner. In the early 1860s Osthaus' parents and younger siblings moved to Mexico in the employ of the Austrian Archduke Maximilian, whom Napoléon III had installed as Emperor of Mexico while the United States was distracted from foreign affairs by the Civil War; when his rule there was overthrown the Osthauses moved to Oshkosh, WI. Edmund joined them in 1883, where he shared a studio with his sister, Marie, also an artist. Shortly afterward he moved to Toledo, OH, at the invitation of David R. Locke, a local newspaperman and collector. He became chief instructor at the and complexity, from the single dog portraits on a typical twenty-four by thirty-six inch canvas to examples of impressive sizes. S. Murray Mitchell's circa 1900 portrait of ten setters, which was six feet high and sixteen feet long and divided into three panels, and was described in the Forest and Stream edition of 28 April 1900. Oshthaus also produced a series of postcards, lithographs and calendar pictures for duPont, including every national champion from the first, Count Gladstone IV in 1896, through Monora in 1911; all were setter dogs except the 1909 winner, the pointer dog Manitoba Rap. At the persuasion of William Bruette, the editor of Forest and Stream magazine, Osthaus executed a series of eleven etchings of subjects including fox hounds, German shepherds, collies, bird dog puppies, a setter bitch and pups, as well as pointers and setters, with a limited number of impressions produced that were signed in pencil by the artist.
Osthaus was a member of the Tile Club in Toledo. He exhibited there regularly, showing such works as his Partridge Shooting and Retriever, both in 1903. At the Art Institute of Chicago (IL) he showed Still Evening in 1903 and, in 1911, Early Rambles and Setters. The Port Huron (MI) Museum of Arts and History has his Major, a portrait of a St. Bernard, and In the Field. The Toledo Museum of Art has another portrait of Major, among other works. The Butler Institute of American Art in Youngstown, OH, has his Family Portrait and his Setters in a Field is at the Morris Museum of Arts and Sciences in Bernardsville, NJ. The Pebble Hill Plantation Museum in Thomasville, GA, has his oils, A Setter and a Pointer and Setters on Point. Other institutions holding his work include the National Sporting Library in Middleburg, VA; the Albany (GA) Museum of Art; and the American Kennel Club Museum of the Dog in St. Louis, MO.
Osthaus died at his hunting lodge in Marianna, FL, on 30 January 1928.