Alexander was born in Ottumwa, Iowa, to Price and Mary Hamilton Alexander, part of a small African American community. He was the eldest of their nine children. When the family moved to a farm outside Des Moines, Price became head custodian at the Des Moines National Bank. Alexander graduated from Oak Park High School in 1905. He then attended Highland Park College and Cummins Art College before matriculating at the State University of Iowa (later known as the University of Iowa) to study engineering. Not only was Alexander was the only black student at the University at the time, but he was the first African American student to graduate from the University of Iowa's engineering program in 1912. His professors warned Alexander that it would be difficult for him to find work as an African American engineer. Alexander was also the first African American football player at the University of Iowa, where he was a three-year starting tackle and earned the nickname "Alexander the Great." Throughout college, Alexander worked multiple part-time jobs to support himself and pay tuition. Alexander was also a member of the predominantly black Kappa Alpha Psi fraternity. During the summer, Alexander worked as a draftsman for Marsh Engineering Company, a Des Moines company that designed many significant bridges which survive even today. In 1921, Alexander also studied bridge design at the University of London while on a sabbatical. He later obtained his civil engineering degree from Iowa State University in 1925.