He attended Miami Northwestern Senior High School, where he played football and also lettered in track.

[1] In 1973, he became the starter at running back after Nat Moore sprained his foot and fractured his left tibia, but would lose playing time with a dislocated wrist that kept him out until the month of November.

He was named the starter at right halfback as a sophomore, but suffered a hip pointer injury and was relegated to a reserve role behind Tony Green, while rushing for 418 yards.

He played in four bowl games and graduated with a bachelor's degree Brinson was signed as an undrafted free agent by the Dallas Cowboys after the 1977 NFL draft and made the team despite fracturing a cheekbone during training camp.

His best game came in the 37–10 win against the Washington Redskins, where he ran for 51 yards on 4 carries, including a 39-yard touchdown run.