He was educated at Dr Butters' College in Shropshire, and spent some time in Germany, before emigrating to Western Australia on board the Protector in February 1830.
His wife died in June 1841, and the following February he married Elizabeth Burke née Tynan.
Over time he became one of the leading merchants in Fremantle, becoming an agent for Lloyd's of London and a representative of the Melbourne Shipowners' Association.
He was a regular contributor to the Fremantle Herald, and became Chairman of the Marine Survey Board.
On 29 July 1872, Bickley was nominated to the Western Australian Legislative Council, remaining in the seat until his death at Fremantle on 30 June 1876.