Born Heinrich von Kleinbach in Berlin, Germany in 1912, his parents emigrated to the United States while he was still an infant. After attending Stanford University, where he was a member of the Alpha Sigma Phi fraternity, he trained as a theatre actor at the Pasadena Community Playhouse and subsequently performed on Broadway, continuing to return to the stage periodically throughout his career.

He made his motion picture debut in 1932 as an uncredited spectator at the Colosseum in The Sign of the Cross. At age 22 in 1934, he played the role of "Silas Barnaby", the villain in the Stan Laurel and Oliver Hardy classic Babes in Toyland. In 1936, having until then been performing under his real name of Henry Kleinbach, he adopted the stage name of "Henry Brandon".

He portrayed the villainous manager of an opera company in Our Gang Follies of 1938. He played the character of "Renouf", a deserter from the French Foreign Legion, in the 1939 remake of Beau Geste. In 1940 he featured in the title role of the successful Republic serial Drums of Fu Manchu. In 1943 he played "Major Ruck", a British Secret Agent in the guise of an SS officer in Edge of Darkness. Kleinbach, standing at 6 ft 4" in height, managed to make Errol Flynn look short in the scenes in which they appeared together in Edge of Darkness in spite of Flynn's height of 6 ft 2". In 1948 he appeared as Giles de Rais in Joan of Arc.

Brandon often played non-European characters, especially Native Americans in Westerns. He also played the Chinese villain Fu Manchu. The successful serial Drums of Fu Manchu (1940) with Kleinbach in the title role was cancelled by its Producer Republic Pictures at the express request of the State Department in 1941 after the USA's entry into the World War 2 out of concern that it was inciting anti-Chinese sentiment in the American public, which conflicted both with the interests of the Chinese-American population and the international relationship with China as an allied power in the war against Japan.

He appeared as the African tribal chieftain "M'Tara" in Tarzan and the She-Devil (1953). In 1956 he played the chief villain, a Comanche chieftain called "Scar", in John Ford's The Searchers. In 1960 he played a Native American character again as "Running Wolf" in the episode "Gold Seeker" in the television series The Rebel. He played Oriental characters in two 1961 episodes, viz. "Angel of Death", and "The Assassins", of the ABC television series Adventures in Paradise. In 1961 he played an American Indian chieftain again in John Ford's Two Rode Together. In 1965, he played the Russian spy Derrick in the two-part episode about Coldfinger on the TV sitcom Mister Ed.

He played a French army captain in Vera Cruz (1954). In 1955 he portrayed Nate Champion, the first casualty of the Johnson County War of Wyoming, in the television series Stories of the Century. He portrayed Jesse James in Hell's Crossroads (1957). In 1958 he portrayed "Acacius Page" in Auntie Mame. In 1959, he played the role of Gator Joe in "Woman in the River" in the ABC/Warner Brothers crime drama, Bourbon Street Beat.

On October 12, 1959 he played the role of Jason in Euripides' Medea as a part of the Play of the Week television series.

Kleinbach lived in West Hollywood in his final years. He died on 15 February 1990 at age 77 of a heart attack at Cedars-Sinai Hospital in Los Angeles. His body was cremated, and its ashes were reportedly scattered at an undisclosed theatre location.