Payne was born in Omaha, Nebraska, to Peggy and George Payne, restaurant owners. He is the youngest of three sons and grew up in the Dundee neighborhood. He is of Greek ancestry. Payne's paternal grandfather, Nicholas "Nick" Payne, anglicized the last name from "Papadopoulos". His family comes from three areas in Greece: the island of Syros, Livadia, and Aegio. Payne's family was part of the fabric of Omaha, which he refers to as part of his upbringing. His grandfather was a founder of The Virginia Cafe, with Payne's father taking over the restaurant. Payne went there regularly as a child. The restaurant was destroyed in a fire in 1969; the W. Dale Clark Library was later built on the site. Payne's paternal grandmother, Clara Payne (née Hoffman), was from a German Nebraska family from Lincoln, Nebraska.
In Omaha, Payne attended Brownell-Talbot School, Dundee Elementary School, and Lewis and Clark Junior High. He graduated from Creighton Prep for high school in 1979. At Prep, Payne wrote a humor column for his high school newspaper and was the editor of the high school yearbook. Payne then attended Stanford University, where he majored in Spanish and History. As a part of his Spanish degree, he studied at Spain's University of Salamanca. He later lived a few months in Medellin, Colombia, where he published an article about social changes between 1900 and 1930. Payne received his MFA in 1990 from the UCLA Film School.
In the 1960s, Payne's father received a Super 8mm projector from Kraft Foods as a loyalty reward, and eventually passed it on to his son when Alexander was about 14 years old.