After a standout career for the Arkansas City High School Bulldogs, Welch helped lead Butler County Community College to an 80-27 record as a starting outfielder. Welch played for the Grizzlies in the 1992 and 1993 seasons. The 1992 team posted a school best record of 48-11.
As a freshman, Brandon played left field and batted .374 from the leadoff spot. His sophomore year, Welch moved to right field and dropped down to the three hole in the lineup where he hit .403. After the 1993 season, Welch was named to the All Jayhawk West and All Region 6 teams and was drafted in the 23rd round by the New York Mets.
From Butler, he went on to have an outstanding two year career at Texas Tech University. As a junior, he hit .374 and was named All-Southwest Conference, All Southwest Conference Tournament and Honorable Mention All-American. That year, he batted cleanup primarily as the DH, but also played some outfield.
His senior year, he batted fifth and played right field and first base and hit .313, again being named All Southwest Conference. That year, the Red Raiders were both Southwest Conference Champions and Southwest Conference Tournament Champions. Tech was ranked number 1 in the nation and lost to Stanford in the NCAA Regional Final.
Following his college career, Welch was drafted by the Oakland A's in the 42nd round. He played one season in the Northwest League with the Southern Oregon A's in Medford. After that season, Welch decided to end his baseball career to enter mortuary school at the Dallas Institute of Funeral Service.
Brandon returned to Ark City where he owns and operates Rindt-Erdman Funeral Home. He married Jennifer Snowden in June, 1995 and they have three children, Lauren, Parker and Gabriel. Welch's brothers were also outstanding baseball players. Jarrett Welch also played at Butler and went on to Emporia State. Tanner Welch played at Wichita State.