Born in Golconda, Illinois, Raum attended the common schools. He later studied law. He was admitted to the bar in 1853 and practiced in Golconda 1853-1856. He moved to Kansas in 1856 and practiced his profession for two years. He then returned to Illinois and settled in Harrisburg.
Following the outbreak of the Civil War, Raum enlisted in September 1861 in the Union Army as the major of the 56th Illinois Infantry, and rose to its colonelcy in 1862. He and his regiment served under Maj. Gen. William S. Rosecrans during the 1862 Siege of Corinth, Mississippi. There, he ordered and led the charge that broke the Confederate left and captured an artillery battery. In 1863, he was part of the army of Ulysses S. Grant that invested and forced the surrender of Vicksburg, Mississippi. He led the 2nd Brigade, 2nd Division, XVII Corps during the Chattanooga Campaign, and was wounded at the Battle of Missionary Ridge in November 1863.
During the 1864 Atlanta Campaign, Raum's brigade held the line of communication from Dalton to Acworth and from Kingston to Rome. He was brevetted as a brigadier general of volunteers on September 19, 1864. In October of that year, he reinforced Resaca, Georgia, and held it against General John B. Hood. Raum was promoted to the full rank brigadier general on February 15, 1865, and served through the end of the hostilities. He led the 2nd Brigade, 3rd Division, XV Corps during the Carolinas Campaign in early 1865.
Raum resigned his commission on May 6, 1865, and returned home to Illinois. In 1866 he obtained a charter for the Cairo and Vincennes Railroad Company, aided in securing its construction, and became its first president. He was elected as a Republican to the Fortieth Congress (March 4, 1867 -March 3, 1869). He was an unsuccessful candidate for reelection in 1868 to the Forty-first Congress.
He then served as United States Commissioner of Internal Revenue from 1876 to 1883. He was acting chairman of the 1880 Republican National Convention in Chicago, Illinois. As United States Commissioner of Pensions from 1889 to 1893, he was responsible for overseeing the acceptance or rejection of pension applications for thousands of Civil War veterans. He engaged in the practice of law in Chicago until his death there on December 18, 1909. He was interred in Arlington National Cemetery in Fort Myer, Virginia.