Cleve was born in Stockholm, Sweden, as the thirteenth child of his father, a merchant known as Fredrik Theodor Cleve. Cleves ancestors on his fathers side came from western Germany and settled in Sweden in the late 18th century.
Cleve showed interest in natural science and natural history from an early age. He attended the Stockholms Lyceum in 1858, studying chemistry and biology. He gained a Bachelor of Science degree from the University of Uppsala in 1863 and a PhD from the same university in 1868.