Rhome attended Sunset High School in Texas, where he played football for his father Byron Rhome. He also practiced basketball and baseball. He accepted a football scholarship from Southern Methodist University. He became a starter at quarterback as a sophomore, registering 74 completions out of 129 attempts for 693 yards, one touchdown and 6 interceptions. In 1961, he transferred to the University of Tulsa after a 2-7-1 football season, and hence the firing of Head CoachBill Meek. In 1963 after sitting out a year, Rhome starred at Tulsa, passing for 1,909 yards, 10 touchdowns, and suffering 13 interceptions. He improved as a senior, with 2,870 passing yards, 32 touchdowns and just 4 interceptions, while leading the nation in total offense. He finished second in the voting for the Heisman Trophy -- to John Huarte of Notre Dame -- in one of the closest elections ever recorded. Rhome held a number of NCAA records, including the largest number of touchdowns in a game, in a season, and the most consecutive passes without an interception in a football season and in his career. Rhome also won the Sammy Baugh Trophy in 1964, and had his Number 17 jersey retired by the school. In 1984, he was inducted into the University of Tulsa Athletic Hall of Fame. In 1988, he was inducted into the College Football Hall of Fame. He also was inducted into the Oklahoma Sports Hall of Fame and the Texas High School Hall of Fame.
Knowing that he had one more year of eligibility, the Dallas Cowboys selected him in the thirteenth round of the 1964 NFL Draft with a future draft pick, as would the New York Jets in the 25th round of the 1964 AFL Draft. Rhome began his professional career in 1965 as the third quarterback behind Don Meredith and Craig Morton; he got a chance to start one game that same year against the Cleveland Browns. He would serve in a backup role until the start of the 1969 season when Roger Staubach joined the team, and at his request the Cowboys traded Rhome to the Cleveland Browns in exchange for a third round draft choice, later used to select Charlie Waters in the 1970 NFL Draft.
Cleveland Browns
In 1969, although Rhome was a backup to Bill Nelsen, the information he provided about the Cowboys offense helped the Browns beat them in a 38-14 Conference Championship playoff win. On January 27, 1970, he was traded to the Houston Oilers in exchange for a third round draft choice, that they gave back to the Dallas Cowboys in payment for their original trade.
Houston Oilers
Rhome would only play as a backup for the Houston Oilers for a year. On August 17, 1971, he was waived after the team decided to keep rookies Dan Pastorini and Lynn Dickey, to back up starter Charley Johnson.
Los Angeles Rams
On August 19, 1971, he was claimed off the waiver wire by the Los Angeles Rams, where he was a backup for one year. On August 24, 1972, he was released after the Rams acquired quarterback Pete Beathard.
Montreal Alouettes (CFL)
In October 1972, he signed with the Montreal Alouettes of the Canadian Football League. On December 18, he announced his retirement.
