Baynham attended North Augusta High School. After playing in only two games as a junior, he had an excellent senior season, that earned him the opportunity to play in the South Carolina All-Star game, where he suffered a knee injury.
As a freshman at Georgia Tech, besides dealing with his knee rehab, he also broke his arm playing against Clemson University. As a sophomore during the fall, he injured the same knee and required surgery.
Baynham enjoyed a healthy season as a junior, setting a school record with 7 receiving touchdowns in a season, while playing wingback, and making 30 catches. He suffered a separated shoulder during his last year in college, that ended up limiting his playing efficiency.
He was a captain of the track team and set a school record with 9.6 seconds in the 100-yard dash.
Baynham was selected by the Dallas Cowboys in the twelfth round (176th overall) of the 1966 NFL Draft and also in the ninth round (73rd overall) of the 1966 AFL Redshirt Draft by the Miami Dolphins of the American Football League. He was selected with a future pick, which allowed the team to draft him before his college eligibility was over. As a rookie in 1967, because of his speed, he was listed as a split end behind Bob Hayes, but also served as a backup at running back. He averaged 27.6 yards on his 12 kickoff returns. He replaced an injured Dan Reeves in the Eastern Conference Championship game against the Cleveland Browns, scoring 3 touchdowns (2 rushing and one receiving).
In 1968, Reeves was the starter at halfback, until week 4 when he tore ligaments in his right knee and was lost for the year. Baynham took over starting 8 games and had his best season in the NFL, registering 438 rushing yards, 29 receptions (third on the team) for 380 yards and 8 touchdowns. He averaged 25.7 yards on his 23 kickoff returns.
In 1969, with Reeves struggling after having off-season knee surgery and Baynham suffering from bruised ribs, it opened the door for rookie Calvin Hill to win the starting halfback job in the second preseason game.
On January 26, 1970, he was traded to the Chicago Bears along with cornerback Phil Clark, in exchange for a second round draft choice (used to draft Bob Asher).
In 1970, he played in 5 games before dislocating a tendon in his ankle. On August 17, 1971, he broke a left wrist in practice that sidelined him for a month. On September 13, Baynham was waived injured after suffering torn ligaments in his right shoulder during a practice.
On May 3, 1972 after a year out of football, the St. Louis Cardinals claimed him on waivers from the Chicago Bears, but would miss half the season with a shoulder injury. On June 9, 1973, he announced his retirement.
His son, Grant Baynham, played for the Georgia Tech as well, receiving the Robert Cup, an award given to school's top student athlete.