He studied philosophy and law in Vienna. After finishing his studies he started further studies at the Bergakademie (Mining Academy) in Schemnitz (Selmecbánya), Lower Hungary (today Banská Štiavnica, Slovakia) in 1763. He studied mining, mechanics, mineralogy and chemistry and after graduating he became a Markscheider (official mine surveyor) in 1768. In 1770 he joined the Hofcommission für die Regulierung der Banater Berg und Hüttenwerke (royal commission for mining in the Banat) where he gained a lot of knowledge on mining in the Banat. He was promoted to the rank of an Oberbergmeister (senior mining official) and became a mine manager in the same year.
In 1775 he became Oberbergmeister in the Tyrol town of Schwaz. Schwaz was one of the largest centres of silver and copper mining in Austria-Hungary in that time.  In 1778 he discovered an occurrence of tourmaline in the Zillertal.
Müller became one of seven Thesaurariats councillor in Transylvania responsible for the supervision of all mining and coinage activities in Transylvania in 1778. After the dissolution of the Thesaurariat Müller became Oberinspector (chief surveyor) of all mining, smelting and salt production in Transylvania.