Brockson was born in Blackbird Hundred, New Castle County, Delaware. He graduated from the Wilmington Conference Academy at Dover in 1890.

Brockson began his career as a teacher and principal in the public schools at Port Penn and Marshallton, Delaware. He studied the law and graduated from the law department of Washington and Lee University in Lexington, Virginia in 1896, when he was admitted to the Bar, and commenced practice in Wilmington, Delaware. In 1908 Brockson was elected to the State House for the 1909/10 session.

Brockson was elected to the U.S. House of Representatives in 1912. During this term, he served in the Democratic majority in the 63rd Congress. He was a supporter during that time of corporal punishment. Seeking reelection in 1914, he lost to Republican Thomas W. Miller, a lawyer from Wilmington, and son of the incumbent Governor Charles R. Miller. Brockson served from March 4, 1913, until March 3, 1915, during the administration of U.S. President Woodrow Wilson. He then resumed the practice of law in Clayton and Wilmington.

Brockson died at Clayton on March 16, 1942, and is buried in the Odd Fellows Cemetery at Smyrna, Delaware.