The soul does go to the afterlife in Greek mythology, just not "heaven" in the Christian sense.
Another thing to keep in mind with Greek and Roman religion is that there was not really "doctrine". Everyone followed the same basic set of beliefs, but there was lots of wiggle room. Three different writers could have three different takes on the same story. New ideas and deities could be added easily with little to no upset, as happened in Rome with Isis, Cybele, and Mithras. Christianity's downfall, arguably, was its insistence on exclusivity. If they could have followed Jesus and still made sacrifices to the emperor and participated in rites of the traditional Roman religion, they might have been accepted.