The Greek afterlife was not in the sky, though. The Greek afterlife was an underworld. Even the Elysian Fields where the blessed dead went (the closest thing to Heaven in Greek mythology/religion) were down below in the realm of the god Hades. The sky was the exclusive realm of the gods, at least certain of them.
That said, the idea of a separate, immortal spirit being contained in a mortal, temporary material body and then released at death is very Greek. As is the idea that the mortal body is corrupt and relatively unimportant compared to that immortal spirit. In fact, some Greek schools of thought went to reincarnation, with the immortal soul returning to a new material body after a sojourn in Hades. This seems to date back to Pythagoras though is most developed in Plato and later. Virgil used it quite effectively as a literary device in The Aeneid. Book 6, where Aeneas visits the underworld, presents a pageant of souls on their way back to begin new lives.