In terms of the afterlife, early Christianity was totally within the context of some Jewish thought of that time. (As regards to the Messiah, dietary laws, the Trinity vs One God, and other aspects of Judaism that clearly differ from Christianity in current belief, the diverse thoughts began early; regarding the afterlife, Christianity closely followed one of the threads of Jewish thought.)
Most of the beliefs of Christianity regarding the afterlife can be found in Jewish works in the time period 200 BCE - 100 CE. Since those works did not make it into the Jewish canon, as closed at Jamnia around 100 CE, they do not appear in the Jewish Bible (or in the Protestant Old Testament).
The concept of a Hell in which sinners are tormented eternally (usually by fire) appears in the (apocryphal to Jews and Christians, alike) Assumption of Moses and Apocaplypse of Baruch, and in 3 Esdras. (3 Esdras is a name given to a Greek copy of Chronicles, Ezra, and Nehemiah that includes additional stories and verses not found in the canonical originals.) Much of the imagery that Jesus used is found in those books. The valley of Gehenna to the south of Jerusalem was a refuse pit, but it also, at one time or another, was the site of the worship of Molech that was supposed to include human sacrifice (2 Kings 23:10, 2 Chron. 28:3). Because of its unholy history, Jeremiah cursed it and the passage in Is 66:24 is often taken to mean that the bodies of the rebels against God (real and figurative) will be thrown there for everlasting torment.
Building on the curses of Jeremiah and the prediction of Isaiah, the later authors of Jewish religious texts symbolically linked the smoldering trash fires of the valley to the ongoing and everlasting torments awaiting those who died in sin.
The imagery that appears on the lips of Jesus and the writings of Paul regarding hell and damnation very closely parallels the imagery that is portrayed in the books I mentioned, above.
In the same way, nearly all the early Christian references to both the resurrection of the saved and to heaven have foreshadowings or parallels in the Jewish apaocrypha (and in the books of the Maccabees that the Catholic church accepts as Scripture). The concept of Paradise (a garden to which the holy will return at the end of time to bring full circle the idea of beginning in the garden of Eden) was promulgated and expanded upon in the apocalyptic books 4 Esdras, The Secrets of Enoch, 2 Baruch, and 3 Baruch.
In other words, the mesage and imagery of Jesus did not attract followers because it was a totally new interpretation of Jewish belief, but because it followed and expanded upon one of the sets of belief common among some Jews in the first century.
On the other hand, the very rise of Christianity probably killed those lines of thought within Judaism. Especially after the early Christians began to see Jesus as man-and-God and create a theology of the Trinity, devout Jews recoiled from this blasphemy. The conflict between Judaism and Christianity became heated. One result was the formal decision of Judaism to close the canon and declare, finally, which sacred writings were and were not truly Scripture. The Jews did not simply say “If Christians use it, we reject it.” However, they did consider characteristics (an age of 500 years as they reckoned it, an origin in the Hebrew tongue to preclude outside influence, along with a consistence of expression throughout all Jewish Scripture) that placed most of the predictions and beliefs that Christians were citing outside the canon.
Could various aspects of Heaven and Hell found in Greek or Roman belief have influenced early Christian thoughts on the matter? Certainly. On the other hand, the basic concepts were pretty well established in Christian thought before Christianity got sufficiently settled into the hellenistic Roman Empire to begin borrowing theology. Descriptions were probably borrowed from the gentiles while the whys and hows were probably based on the older Jewish concepts.