A St. Louis native, Christman led the Missouri Tigers to a 20–8 record during his three seasons as their starting quarterback. He was a two-time All-American, and led the nation in touchdown passes in 1940. He was Missouri's all-time leading passer until 1976, when he was surpassed by Steve Pisarkiewicz. While at the University of Missouri, he was a member of the Kappa Sigma fraternity. His jersey number, 44, is one of seven retired by the school. In 1956, he was inducted into the College Football Hall of Fame.
Christman played six seasons in the National Football League, from 1945 to 1950. He was a member of the so-called "Dream Backfield", which led the Chicago Cardinals to the 1947NFL Championship. A notoriously poor ball-handler, at one time he owned the record for most fumbles in a game (five) and most own fumbles recovered in a season (eight).
After retiring as a player, Christman worked as a television color commentator, first teaming with play-by-play announcer Joe Boland to call Cardinals games for CBS in 1958 and 1959. In 1962, he began calling American Football League games on ABC with Curt Gowdy, a pairing that continued after AFL rights shifted to NBC in 1965. Christman called Super Bowl I with Gowdy for NBC in January 1967. In 1968–69 he returned to CBS, teaming with Ray Scott on NFL broadcasts.
Christman also called the collegiate Orange Bowl game for several years, teaming with Boland (1960), Scott (1961), and Gowdy (1962–67). He and Gowdy then called the Rose Bowl game in 1968.