Born in Lancaster, Garrard County, Kentucky, Bradley was educated by private tutors and at a private school in Somerset, Kentucky. He entered the Union Army during the Civil War at the age of fifteen, but only served briefly due to his youth. He studied law and was licensed to practice in 1865, becoming the prosecuting attorney of Garrard County in 1870. In 1889, he was appointed Minister to Korea, but declined the job. From 1890 to 1896, he was a member of the Republican National Committee. He was the Republican governor of Kentucky from 1895 to 1899. His sister's son Edwin P. Morrow was also governor of Kentucky, from 1919 to 1923.

He was elected as a Republican to the U.S. Senate and served from March 4, 1909 until his death in Washington in 1914. During the Sixty-first and Sixty-second Congresses, he was chairman of the U.S. Senate Committee on Expenditures in the Department of Justice. He was also chairman of the Committee to Investigate Trespassers upon Indian Land during the Sixty-first Congress, and the chairman of the Committee on Revolutionary Claims during the Sixty-third Congress. He was buried in the state cemetery of Frankfort, Kentucky.