Kinchen was born in Baton Rouge to parents Gaynell "Gus" Kinchen and Tori Kinchen. He attended and played football at University High School. His father was a member of the 1958 LSU Tigers football team and one of the famed Chinese Bandits.
In 1991, Kinchen signed with the Cleveland Browns where he played for head coach Bill Belichick. Over the next five seasons, he played in all but five regular season games (75 total). His best seasons were from 1993–95 when he caught 73 passes for 795 yards and three touchdowns in 35 starts. 
Kinchen was a member of the inaugural Baltimore Ravens team in 1996 which resulted from the Cleveland Browns relocating to Baltimore. In 1996, he had his best season as a tight end with 55 receptions for 581 yards (both career highs) and one touchdown. In three years with the Ravens, Kinchen totaled 79 catches for 786 yards and two touchdowns.
In his two season with the Carolina Panthers, Kinchen spent most of his time at long snapper and on special teams. As a tight end, he had six catches for 52 yards and one touchdown. After suffering an injury, he was released at the end of the 2000 season.
Kinchen lives in Baton Rouge, where he and his wife Lori have raised their four sons. His brother Todd also played in the NFL. His sons Austin and Hunter Kinchen were long snappers on the LSU football team. Both Brian’s mother Toni and his wife Lori were LSU cheerleaders.
Since retiring from football, Kinchen has spent time as a high school football coach and was a volunteer coach for Nick Saban’s 2004 national championship team at LSU. He currently serves as a college football analyst for ESPN. He was also previously a commentator for ESPNU, but was suspended after making a comment during a game that receivers needed to use their soft hands to "caress" the ball, and then calling the comment "kind of gay."