Clarence was born on February 3, 1849, in Jewett, New York, son of Jason F. Bloodgood (1814–1906) and Lucinda H. Coe (1809–1893).
In the fall of 1871, he began attending Yale University, from which he graduated in 1875 with a degree in Classics and honors.
Among his classmates were Michigan Senator John Patton Jr. and Vermont Governor Edward Curtis Smith.
He began teaching before he started Yale, and in November 1875 he was elected as a Democrat to be school commissioner for the Greene County 1st District.
He was elected to the New York State Senate in 1891, where he represented the 14th District (Greene, Schoharie and Ulster counties).