He was born in Bonn, and played the violin in the court orchestra there before becoming concert master of the orchestra of Prince Heinrich of Prussia. He moved to London in the early 1780s, where he worked as a composer and played violin in a string quartet.
He is perhaps best known today, however, as a concert organiser. He brought Joseph Haydn to London in 1791-72 and 1794-95. Haydn wrote his symphonies numbers 93 to 104 for these trips, which are sometimes known as the Salomon symphonies (they are also known as the London symphonies). Salomon is also believed to have given the Jupiter nickname to Wolfgang Amadeus Mozart's Symphony No. 41.
Salomon died in London.