A native of Alba, Texas, Dawson played collegiate basketball at Paris Junior College, where he attained the nickname, "Big Orange" and Baylor University during the late 1950s and early 1960s. In 1960, the 6'5" center earned All-Southwest Conference honors after averaging 16.4 points per game for the Baylor Bears. After his college career, he was drafted into the Army, stationed in Fort Knox as a tank commander. From 1973 to 1977, he served as Baylor's men's basketball head coach. He then worked as a scout for the NFL's Dallas Cowboys and as a salesman for Converse shoes before becoming an assistant coach for the Houston Rockets in 1979.
In 1989, Dawson was struck by lightning during a golf outing, and over the next few years his vision became progressively worse. Unable to continue his coaching duties, Dawson moved to the Rockets' front office, becoming general manager in 1996. Among his most notable accomplishments were his selection of Yao Ming in the 2002 NBA Draft and his seven-player trade to acquire NBA scoring champ Tracy McGrady in 2004.
After announcing Daryl Morey as his successor, the Rockets honored Dawson by hanging a banner with the initials "CD" in the rafters of the Toyota Center during a game with the Phoenix Suns.