Bach was born in Berlin. He studied under Adam Friedrich Oeser in Leipzig. In May 1773, he moved to Dresden, and in February 1776 he moved to Hamburg, where his father was Director of Music. In September 1776 he embarked on a study trip to Rome, where he became seriously ill soon after his arrival in February 1777, and died of this (unknown) ailment in 1778.
Bach created mostly brush drawings of idyllic landscapes, bustling with people. His works show the influence of Solomon Gessner. Towards the end of his life he turned to representations of people and created historical and mythological scenes. He also made vignettes and illustrations of works by Gottlieb Rabener and Christian Felix Weisse.
In his time he was quite renowned as an artist. Collections of his works are in Coburg, Dresden, Hamburg, Leipzig and Vienna.