Geiogamah was born in Lawton, Oklahoma to a Kiowa father and Delaware mother. He graduated from Anadarko High School and studied journalism at the University of Oklahoma. He later attended Indiana University Bloomington. Following his graduation, he landed a job as the public affairs liaison for Commissioner of Indian Affairs Louis Bruce within the Bureau of Indian Affairs under President Richard Nixon.
In late 1971, Geiogamah formed a theater group at the La MaMa Experimental Theater Club in New York City's Lower East Side. Initially called the American Indian Theatre Ensemble, the group changed its name in 1973 to the Native American Theatre Ensemble. (The reason for this name change "was complex but simple," the company explained in text that appeared in a 1973 show program. "Too many non-Indians who approached us during [our] tours [and] after performances...seemed unable to understand that we were real people, really alive and breathing, and that we were certified residents of the United States of America.") Geiogamah's first play for the company was Body Indian in 1972 followed by Coon Cons Coyote and Foghorn. The group produced his final play 49 in 1982.
In 1980, Geiogamah became the author of New Native American Drama: Three Plays, published by the University of Oklahoma Press.
Geiogamah later formed the widely acclaimed American Indian Dance Theatre, which gave its first public performance in 1987 with Geiogamah as its director and Barbara Schwei as its producer. The 24-member dance troupe represented about 18 Indian nations and toured both nationally and internationally. The dancers wore a variety of traditional costumes, and the music was performed on traditional instruments made by the performers. The group made their New York City debut in 1989 in Manhattan's Joyce Theater.
In 1990, the group was featured in PBS' Great Performances in the segment "The American Indian Dance Theater: Finding the Circle". The New York Times praised the group saying that the "hallmark of this company is its authenticity" with "serious artists conveying basic facts of their lives and cultures." In 1993, the Dance Theater was produced as a segment for Dances for the New Generations for the PBS television series Great Performances/Dance in America. Barbara Schwei and Hanay Geiogamah were producers and Phil Lucas and Geiogamah were directors.
Geiogamah served as producer and co-producer for the TBS multimedia project, The Native Americans: Behind the Legends, Beyond the Myths aired on TNT from 1993 to 1996. The program was a series of fact-based historical dramas and publications. Geiogamah was co-producer on "The Broken Chain", which told the story of the Iroquois Confederacy during colonial times and also for "Geronimo" (executive produced by Norman Jewison). In 1994, he was co-producer for "Lakota Woman: Return to Wounded Knee", and a year later he was again co-producer for "Tecumseh", the story of the Shawnee leader who fought against the United States during Tecumseh's War and the War of 1812. Geiogamah returned in 1996 as producer for TNT's "Crazy Horse", the Native American war leader of the Oglala Lakota.
In 2009, Geiogamah was co-executive producer for The Only Good Indian, an independent Western starring Cherokee actor Wes Studi.
In 2010, Geiogamah joined co-host Robert Osborne of Turner Classic Movies for “Race in Hollywood: Native American Images on Film”, a series that looked at both positive and negative images of the Hollywood Indian.
Geiogamah serves on the National Film Preservation Board established in 1988 as an advisory body to the Librarian of Congress' National Film Registry.
From 2002 to 2009, Geiogamah served as the director of the UCLA American Indian Studies Center and was a founder and co-director of "Project HOOP" (Honoring Our Origins and Peoples), a national, multi-disciplinary initiative to establish Native theater in tribal colleges, Native communities, K-12 schools, and mainstream institutions.