Bleekman was born on March 26, 1846, in Salisbury, Herkimer County, New York. During the American Civil War, he served with the 2nd Ohio Cavalry of the Union Army. Battles he took part in include the Battle of Hanover Court House, the Battle of Malvern Hill, the Battle of the Wilderness, the Battle of Spotsylvania Court House and the entirety of Wilson's Raid. Bleekman mustered out of service in 1865 after suffering a severe leg injury. Afterwards, he attended Albert College before moving to Tomah, Wisconsin in 1869.
Bleekman married twice. He and his first wife, Eliza, had one son before her death. Bleekman and his second wife, Alice, would have three children. He died on June 27, 1908, in La Crosse, Wisconsin.
Bleekman was elected to the Assembly in 1872 before representing the 4th district of the Senate from 1874 to 1875. Later, he was District Attorney of Monroe County, Wisconsin from 1876 to 1878. Bleekman was a Republican.