This is a harder question than you might think. The fact is that in the Old Testament the idea of heaven and hell is not fully developed. This is an example of what is sometimes called "progressive revelation." God fully revealed his nature gradually, over time, to his people. There are a number of concepts only introduced, and sometimes not very clearly in the OT, which are fully revealed in the New Testament. For example, God’s intention with regard to divorce is only fully revealed in the New Testament. The idea of grace and the implications of the blood sacrifice system is foreshadowed in the Old Testament, but fully realized in the New. A good example of this principle is the biblical teaching on what happens after we die. The idea of heaven and hell is not established with much specificity in the Old Testament. David talks about Sheol and Abaddon in the Psalms. It is only in the book of Daniel, in the Old Testament, that the idea of a final resurrection is fully revealed. There are many passages in the Old Testament which hint at a Judgement Day, at the afterlife and heaven and hell, but the honest truth, at least as I see it, is that these ideas are suggested but not clearly stated. Of course, in Daniel 12, the final judgement, heaven and hell are described quite clearly.
You are probably aware that the Pharisees and Saducees argued over the nature of the resurrection (Acts 23:5-10). This is because the Saducees only used the first five books (the Pentateuch). If you only read the Pentateuch, you will not find the resurrection ever taught, at least in an unmistakeable way. The Pharisees accepted Daniel as scripture and therefore agreed with Jesus and Paul about the resurrection.
The same can be said about the nature of Hell. The Old Testament clearly reveals Satan and his work, but it only somewhat vaguely tells us about the nature of Hell. David talks about Abaddon, which is the waiting place of those who rebelled against God, and Daniel implies that those who are not righteous will suffer "everlasting contempt." (Daniel 12:2). So we have the doctrine of Hell in the Old Testament, but it is not a major topic. On the other hand, clearly final judgement, heaven and hell are quite thorougly worked out in the New Testament.
Here is where the skeptical, unbelieving scholars step in. They point out that the Greeks had a teaching about an underworld, called Tartarus. I believe it is fair for scholars to say that the Greeks had a more thoroughly worked out idea of the underworld/afterlife than the Jews did in the centuries before Jesus lived. The Greek idea of Tartarus would be very roughly equivalent to the Jewish/OT teaching on Abaddon (the waiting place), not Hell (the place of final judgement). We cannot completely blame the skeptical scholars for proposing that the Jews borrowed their ideas about judgment, Hades and Hell from the Greeks. As a academic theory it is not unreasonable. It is true that the Jewish idea of the afterlife was somewhat vague, that the Greek idea was more well developed and that the New Testament teaching, like the Greek one, is more developed. However, we, as Christians, have a different perspective on this. We understand that "All scripture is inspired by God." We understand that the Bible writers got their ideas, not from Greek mythology, but by inspiration from God.
I believe that the data we have is more consistent with the belief that the Jewish idea of hell comes from the Old Testament and from direct revelation to Jesus and the apostles, not from Greek mythology. First of all, these scholars greatly exaggerate their case in order to "prove" their point. They are not correct when they say that the Old Testament did not include a doctrine of the afterlife, heaven and hell. Second, the New Testament idea of judgment, heaven and hell is far more similar to the Old Testament version, no matter how vague, than anything in Greek mythology. Both the OT and the NT idea of judgment, heaven and hell is monotheistic, for one thing. Also, the parallels between Daniel and Revelation 21-22 are undeniable. The parallels with the Greek idea of crossing the Styx river is pretty much non-existent. This is just one more case of scholars grasping at straws in an attempt to undermine our confidence in the Scripture. I believe that it is FAR more believable, based on the evidence (not even allowing for what we know–which is that the scriptures are inspired) to believe that Jesus and the apostles got their ideas about the afterlife, both from the Old Testament and from the same God who inspired the writing of Pslams, Daniel etc., than it is to believe that it came from the Greeks. Might the actual vocabulary used to describe the afterlife in the Greek language have come from Greek words. Of course, but obviously Greek speakers would use Greek words for these things, but the ideas did not come from the Greeks.
This brings me to the question of inspiration. The extremely biased perspective of the skeptical unbelieving scholars do not allow them to see the obvious, which is that the Christian idea of the afterlife is much more closely related to the OT than the Greek ideas. Let me add to this the question of inspiration. If we will allow for the idea that Jesus was inspired by God (which is pretty obvious, given his resurrection of Lazarus, walking on water, fulfilling the prophecies, his resurrection and a zillion other things), then the idea that he got his idea of the afterlife from Greek mythology becomes really quite absurd. Seriously, this claim is downright absurd. If we look at the life of Jesus, how can anyone reasonably propose that he got his concept of the afterlife from mythology, when it is quite clear that he had direct access to the Father in heaven?
Of course, in a scholarly setting, such things are generally discounted. In that arena this claim that Paul borrowed his teaching on hell from the Greeks is in play. I will have to admit that there is even some evidence that supports this conclusion. However, I believe even if we ignore the obvious, which is that "All scripture is inspired by God," it is still far more reasonable to conclude that the church got its idea of the afterlife principally from Jewish sources and borrowed vocabulary, but not ideas from the Greeks.