Grune was born in Vienna, where he later attended drama school. He volunteered in the First World War, where an injury temporarily deprived him of the ability to speak in 1918.
After the war he made his directing debut in 1919 with Menschen in Ketten ("People in Chains"). In 1923 he made Schlagende Wetter with Liane Haid and Eugen Klöpfer in the leading roles. The film is a notable early example of naturalism in film making, at a time when expressionism was the norm. Also that year he made Die Straße ("The Street"), which considered Grune's most notable film. In 1926 he made Die Brüder Schellenberg ("The Brothers Schellenberg") with Conrad Veidt and Lil Dagover. Many of his early films are now lost.