Müller was born in Bristol, the son of J. S. Müller, a Prussian, curator of a museum in Bristol. He first studied painting under James Baker Pyne. His early pictures were mostly of the scenery of Gloucestershire and Wales, and he learned much from his study of Claude, Ruysdael, and earlier landscape-painters. He witnessed the 1831 Bristol riots and recorded some of the scenes in a series of “raw and brilliant oil and watercolour sketches”.In 1833, he exhibited at the Royal Academy for the first time, showing Destruction of Old London Bridge-Morning. The next year he made a tour through France, Switzerland, and Italy.

His second visit was to Lycia in south west Turkey in 1843-44 when Charles Fellows was removing the Xanthus Marbles for the British Museum. His journey was at the request of the archaeologist Charles Fellows. Müller and his pupil Harry Johnson accompanied the government expedition to Lycia. He spent three months sketching the landscape and local people around Xanthus, Pinara, and Tlos. He spent most of the rest of his life, after his return to England, working on watercolours, and a few oils, of Lycian subjects. The work he carried out at Lycia is considered to be among his finest.
In 1840, he again visited France, where he executed a series of sketches of Renaissance architecture, twenty-five of which were lithographed and published in 1841, in a folio entitled The Age of Francis I. of France.
The British Museum was bequeathed a rich collection of Müller’s sketches by art collector John Henderson.