Coston was the oldest son of Birmingham physician Hamilton Coston and his wife, the former Roseanna Caughran. He completed his undergraduate degree at the University of Alabama in 1916. While in college he was a founding member of the Iris Club, which was chartered later as the Alpha Phi Zeta chapter of Lambda Chi Alpha fraternity, manager of the Glee Club, and a performer with the Blackfriars theater troupe. He was also a member of the King Solomon Lodge No. 460 of the Free & Accepted Masons.
Coston was enrolled at the University of Alabama School of Law when the United States entered the "Great War" in Europe. He volunteered for officer training at Fort McPherson, Georgia and was commissioned as a second lieutenant He spent time at Camp Sheriden, Camp Beauregard, Camp Wheeler and Camp Gordon before sailing for France with the 328th Infantry in April 1917. Over the next 18 months he was involved in several battles and earned promotion to First Lieutenant.
On October 6, 1918, during the conclusive Battle of the Argonne Forest, the 328th began advancing through Chatel-Chéhéry, fighting their way north toward the hillside village of Cornay. On the morning of the 9th, Coston led his platoon in charging a German machine gun nest on a steep hill and was wounded in the leg. He dragged himself on and urged his men forward, but was felled by another shot to the head. Coston was awarded a posthumous Silver Star for his sacrifice. His body was returned to Birmingham and he was buried at Elmwood Cemetery.