Klaproth was born in Wernigerode. He was the son of a tailor, and attended the Latin school at Wernigerode for four years.
For much of his life he followed the profession of  apothecary. In 1759, when he was 16 years old, he apprenticed at Quedlinburg. In 1764, he became a journeyman. He trained in pharmacies at Quedlinburg (1759–1766); Hanover (1766–1768, with August Hermann Brande); Berlin (1768); and Danzig (1770).
In 1771, Klaproth returned to Berlin to work for Valentin Rose the Elder as manager of his business. Following Roses death, Klaproth passed the required examinations to become senior manager.  Following his marriage in 1780, he was able to buy his own establishment, the Apotheke zum Baren. Between 1782 and 1800, Klaproth published 84 papers based on researches carried out in the Apothekes laboratory. His shop was the most productive site of artisanal chemistry investigations in Europe at that time.
Beginning in 1782, he was the assessor of pharmacy for the examining board of the Ober-Collegium Medicum. In 1787 Klaproth was appointed lecturer in chemistry to the Prussian Royal Artillery.
In 1788, Klaproth became an unsalaried member of the Berlin Academy of Sciences. In 1800, he became the salaried director of the Berlin Academy of Sciences.  He sold the apothecary and moved to the academy, where he convinced the university to build a new laboratory. Upon completion in 1802, Klaproth moved the equipment from his apothecary laboratory into the new building.
When the University of Berlin was founded in 1810 he was selected to be the professor of chemistry.
He died in Berlin on New Year's Day in 1817.