The short answer is, early Christian culture was predominantly Greek, and the idea of an underground land of the dead ruled over by a god comes directly from the Greek Hades. (Confusingly, both the god and his domain were named Hades. In turn, the English "Hell" comes from the name of the god of death in pre-Christian Norse mythology.)

A bit more complicated is why Christian Hell is a separate place of torment for bad people. Greek Hades assigned rewards and punishments, and the Vikings had their famous VIP area for warriors, Valhalla, but neither arrangement is quite like Hell. And in Judaism, God doesn't allow naughty souls into the afterlife at all--they're totally destroyed after judgement. The missing piece here is the character of Satan. Even though his name is Hebrew ("Adversary"), his origin is in Zoroastrianism, the religion of ancient Persia--sort of the Fifth Beatle of Western culture.

While Zoroastrianism isn't technically "Abrahamic" (that is, unlike Judaism, Christianity, and Islam, its theology doesn't include the Biblical character of Abraham), the contact between ancient Israel and Persia introduced some key Zoroastrian concepts into Judaism, part of the brewing schism which eventually produced Christianity and Islam. Like Abrahamic religions, Zoroastrianism has one god, Ahura Mazda. But he's split into two incarnations at war with each other: his good aspect is Spenta Mainyu (Jehovah), who lives in Vahishta Ahu (Heaven) with an army of yazatas (angels). His evil aspect is Angra Mainyu (Satan), who lives in Achista Ahu (Hell) with an army of devas (devils). Spenta Mainyu and Angra Mainyu will battle over the world until the coming of the Saoshyant (Messiah), who will make peace between them.

The last standardized version of the Hebrew Torah was written around 400 BCE; the Greek Septuagint (the proto-Christian Old Testament, largely translated from the Torah but with some bits changed) was written around 200 BCE. And the Latin Vulgate translation of the Septuagint, the Catholic Old Testament, was written in 382 CE. Comparing the differences between the three neatly illustrates the expansion of Zoroastrian ideas in Christianity and Judaism during the 1000-year gap. For example, the Torah version of the Book of Job begins with a conversation between God and "a devil." ("Now there was a day when the sons of God [angels] came to present themselves before the Lord, and a devil came also among them.") But the Latin Vulgate version has the conversation taking place between God and "Satan"--no longer an employee but a cosmic rival, if not quite as powerful as his Zoroastrian ancestor.

(Update: As /u/koine_lingua points out, while some changes in the Christian Old Testament were already apparent in its first Greek version, the transformation of "a devil" into the Devil happened centuries later, in the Latin.)

While I can't comment on where it originated, I can say that it did not come from the Bible itself. 2 Peter 2:4 states that the angels that rebelled were sent to Hell and placed "in chains of darkness, to be held for judgment". The idea of Satan ruling over Hell come from later sources.

As /u/N1ckFG pointed out, much earlier than Dante, we can see the precedents of this idea in early Greek thought (e.g. Hades/Pluto), which would enter into Jewish/Christian afterlife mythology and evolve from there (cf. Bauckham 1998:224). In my comment here, I touched on that a bit more.

A nice starting place to look for some early hints of sort of "bridge" between the two would be towards the motifs found in the book of Enoch, where angels are assigned specific functions in overlooking various aspects and areas of the cosmos. Enoch and related writings had a significant impact on other Jewish thought of the time; and it also influenced several Christian texts from around the 2nd-3rd century CE, which were fairly popular at the time: the Apocalypse of Peter and the Apocalypse of Paul (even appearing in the earliest list of "canonical" [or quasi-canonical] Biblical books). These would have a lasting influence, with the former possibly being the most influential text that would lead to our picture of the Dante-esque "Hell."

In Apocalypse of Peter, the angel in charge of Hellish punishment is called Tartarouchos. But in other texts, the name of this angel is going to shift: for example, it's going to be Belial/Beliar (e.g. the Testament of Dan); and this figure is closely associated with Satan.