For once, one of these notes is relatively easy. Many polytheistic/pagan faiths have a belief in some sort of afterlife which eventually cycles around to the soul being reborn into a new life. Some people tend to look at Hellenic polytheism as being essentially different in that reincarnation does not happen in it (as far as they are concerned). Personally, I find that way of thinking to be absolutely pure bullshit. The dead in both the Plains of Asphodel and in the Elysian Fields can CHOOSE to be reborn as can the souls who are being punished in Erebus and Tartarus after they have paid the price for their sins and repented of them (there are a few exceptions to that rule for people like Sisyphus and Tantalus). This is verified in both the mythology AND the philosophy of ancient Greece.
In the mythology, there are two springs that the soul can drink from before being reborn. Most drink from the spring of Lethe (forgetfulness) and therefore have no memories of their previous lifetimes. The wiser ones drink from the other spring, Mneme (Μνημη) — the spring of Memory — so as to carry their memories with them. There is also the part of the mythos which indicates that if the soul earns the Elysian Fields three times (need to be reborn to do this, obviously) that they attain the Isles of the Blessed. The fact that these stories exist is evidence of reincarnation as a belief found in Hellenismos.
HOW you get to the Elysian Fields is pretty much a simple matter. It is not (despite popular conceptions) reserved only for those who are ‘Heros’ in the narrow sense of the word. Everyone can live a life worthy of them, by actually actively helping others. BTW, Christianity actually has a good set of guidelines for what constitutes actively helping others.
I was hungry and you gave me food, I was thirsty and you gave me drink, I was a stranger and you welcomed me, I was naked and you clothed me, I was sick and you visited me, I was in prison and you came to me
THESE actions when taken by anyone, can fit the broader definition of a ‘hero’ and they can ‘qualify’ you for the Elysian Fields. However they must be something that is consciously done NOT for the express purpose of ‘a good afterlife’, but simply because we share in a common humanity with everyone else.
Now to jump into Greek philosophy, Plato in his dialog Phaedo describes a conversation attributed to Socrates that specifically discusses the afterlife and the immortality of the soul. in it Socrates explores various arguments for the soul’s immortality in order to show that there is an afterlife in which the soul will dwell following death. In point of fact Socrates actually uses four different approaches that result in the same conclusions based on his logic.
Now, there is some discussion that has been going on in the modern era about near-death experiences. That is not and should not be a reason in and of itself to adopt beliefs in the immortality of the soul, the existence of an afterlife or reincarnation. At best, they should be a stepping off point. Beyond that, I cannot tell anyone else that they MUST believe in a certain way. To act in that way would be to negate the orthopraxis which is a hallmark of Hellenismos.