Born in Leipzig, Klengel studied with Emil Hegar in his youth. After his 15th birthday, Klengel joined a professional orchestra and began touring Europe and Russia. Klengel also became a soloist at that point, frequently giving solo performances. He soon became principal cellist of the orchestra, where he remained for over four decades. During that time period, Klengel became professor at the Leipzig Conservatory, and began composing. He ultimately composed hundreds of pieces for the cello, including four violoncello concertos and various etudes and technique pieces. Klengel died in October of 1933 in his beloved hometown of Leipzig.
