Born on a farm near Greenville, South Carolina, Ashmore attended the public schools of Greenville. He graduated from Furman University Law School, Greenville, South Carolina, in 1927. While a student he engaged in agricultural work, retail sales, and as a substitute rural mail carrier. He was admitted to the bar in January 1928 and engaged in the practice of law in Greenville, South Carolina. He served as solicitor of Greenville County Court from 1930–1934, and then as a solicitor of the thirteenth judicial circuit of South Carolina from 1936–1953. During World War II, while on official leave from duties as a solicitor, Ashmore volunteered for service in the United States Army in December 1942, serving in the United States and overseas until discharged from active duty in May 1946, as a lieutenant colonel in the United States Army Reserve. He was promoted to colonel in 1955.
He served as member of the board of South Carolina Appalachian Regional Planning and Development Commission (later South Carolina Appalachian Council of Governments) from 1970 to 1989, and chairman from 1970 to 1972. Ashmore was also elected to the Common Cause National Governing Board in 1973. He was a resident of Greenville, South Carolina, until his death there on October 5, 1989. He was interred in White Oak Baptist Church Cemetery, Greenville, South Carolina.