Joseph Wilkins Cleaver was born at Evandale, Tasmania, on 17 October 1883 to parents Thomas and Mary Anne (née Rowbottom) Cleaver. He was one of many siblings. In his early adulthood, Thomas appears to have moved to Victoria. He married Henrietta Maud Phillippi in 1911, and they had two daughters: Nellie Vonde Mary, born in 1912, and Valetta Irene, born in 1916. The family resided in Bendigo, Victoria. Joseph worked as a theatrical manager and picture showman.
Joseph Cleaver enlisted in Australian Imperial Force as a private on 8 May 1916. He lied about his age when he enlisted, saying that he was several years younger than he was in reality. His middle name is not shown on his service record. Cleaver embarked from Melbourne on HMAT Nestor on 2 October 1916.
He was initially assigned to the 59th Australian Infantry Battalion, but was soon transferred to the 60th Australian Infantry Battalion. He served with this battalion in France, acting as a Corporal or Sergeant on many occasions as required. Cleaver was transferred back to the 59th Australian Infantry Battalion in January 1918. On 24 May 1919, Joseph Cleaver was the bandmaster for a performance of the Number 5 Australian Group Regimental Bands on Weymouth Hospitals Day at Alexandra Gardens in Weymouth, England. He also presented a cornet solo of “The Rosary” as part of this show. Cleaver returned to Australia aboard Port Littleton on 10 June 1919.
After the war, Joseph Cleaver enlisted for home service and was assigned to the Military District Australian Army Reserve Band. He served in the band from 13 October 1919 to 25 February 1920. After his discharge, Joseph settled with his family in Melbourne and worked as a musician. Between the mid-1920s and the early 1930s, Joseph and Maud established and managed Cleaver Knitting Mills, which manufactured high quality clothing and outerwear. Their daughter Nellie also assisted in the management of the Mills when she was old enough to do so. Joseph had retired by 1954, and died in 1956 at the age of 73.