Robertson was born in Woodford County, Kentucky, the son of Scottish immigrant Cornelius Robertson and his wife Clarissa Hill (Keech) Robertson. When Robertson was only four years old, his father died, leaving his mother almost penniless. Unable to properly support her family, she apprenticed young Robertson four years later to a hatter, who moved with the boy in 1824 to St. Louis. After studying medicine at Transylvania University in Kentucky, Robertson graduated in 1835. With the Texas Revolution emerging as a national topic, Robertson joined a company of Kentucky volunteers as a lieutenant and made plans to travel to Texas. However, they were delayed in New Orleans and did not arrive in the Republic of Texas until September 1836. There, he joined the Army of Texas and was commissioned as a captain.