The idea of which didn't originate in Ancient Greece but the topic was brought into a better light by some of its most famous philosophers. The earliest Greek who mused on the topic of 'metempsychosis' was Pherecydes, the purported teacher of Pythagoras.
The idea of reincarnation is explored in the Orphic tradition which appeared around 6th century BC in which the mythic figure of Orpheus claims that the soul is divine and unfairly tethered to the prison of the body. After death, the soul and the body are separated but not for long, and it is once again joined with mortal flesh. The soul must live several lives in other to become enlightened and truly divine until it can exist free of the body.
Plato was another famous philosopher who believed this to be true, using one of his works, The Republic, to retell this idea in the guise of an experience/myth. Er, the son of Armenius returned from the dead after twelve days and retells what he saw of the death realm. He tells that he went to the place of judgement and saw other souls there choosing another body, another reincarnation for their soul in order to return to the world of the living, both human and animal. Orpheus, Thamyra and Antalanta were supposedly some of the souls amongst these. They chose there next form and then drank from the river Lethe (oblivion, forgetfulness) and shot away to be reborn.
Plato also believed that the number of souls was finite, so that new ones were not created, every one transmigrated from one body to the next.