Adkins was born on January 16, 1892, in Newport, Ohio, the son of Emily (née Middleswart) and Alvin Adkins. He grew up on a farm with his brother and sister. After attending and finishing high school in Newport, he entered Denison University. Having a reputation as a tall and shy boy, Adkins graduated in three and a half years. Adkins then spent three years at Ohio State University. He took his Master's degree in 1916, and his Ph.D. in 1918, under the direction of William Lloyd Evans. After receiving his degree, he began work as a research chemist for the United States Department of War. In the following academic year, Adkins served as an instructor in organic chemistry at Ohio State University and in the summer of 1919 he was a research chemist with E. I. Du Pont De Nemours and Company.