The Greek underworld (Haides) was neither punishment nor reward, but simply the next state of existence. In earlier Greek literature, it is a dark and smoky and mouldy place (σμερδαλέ᾽ εὐρώεντα "dire and dank," Iliad 20.65) where all the souls of the dead roam. They are technically not in pain, but are generally understood to be in a place that is somehow less than the world above. So in Odyssey 11, Odysseus travels to Haides and comes across the shade of Achilles, who says he would rather be a slave up above in the land of the living rather than the famous Achilles in the lands of the dead. We hear about the dead in the leimôn asphodelon, the fields of asphodel. The asphodel plant was colorless and without distinctive flavor, but edible and good enough in an emergency. That is a great metaphor for the Unseen Lands in Greek thought.

In Hesiod, we hear about the pit of Tartarus, where the Titans were imprisoned. It, unlike Hades, was a place of torment and pain, much more like a Christian Hell, but it was also not a place that mortals could wind up, at least not in Greek thought, unless they were very very naughty (but see below). It is a place for the punishment of gods, as Zeus threatens in Iliad 8.13ff:

"And any one I perceive against the gods' will attempting to go to help Trojans or Greeks, he shall go whipped against his dignity back to Olympos; or seizing [him] I will hurl him down to murky Tartaros (ἠερόεντα--great word), far below, where the uttermost depth of the pit lies under earth..."

Judges exist in the underworld, like Minos (Od 11.567) or Persephone, but the point of the underworld was not generally judgment and retribution except for the very specific circumstances of the Pit of Tartarus. It was also not a place for reward, like Christian heaven. In the Classical period, the Greeks began to modify Homero-Hesiodic notions of the afterlife to include "happy" places like the Isle of the Blessed (and later, the Roman concept of the Elysian Fields). Like Tartarus, this was not a place that mortals could endeavor to go, no matter how goodly they were in life. Like Tartarus, it was a place for special people: heroes, celebrity mortals, and the like. Also in the Classical period are the emergence of mystery cults, which among other things promised their adherents a happier alternative to the dreary afterlife of Greek thought. The Mysteries at Eleusis are the best example of this, but there were others, like the Orphic Mysteries. See here. Christianity was originally just another mystery cult, albeit one which was ultimately much more successful in gaining followers than the others.

There is a lot of inconsistency in Greek thought about the general state of the dead in the underworld. In Homer we hear that the dead, having passed beyond the river Lethe, forget about their former selves:

[Achilles talking of Patroklos]: "Though the dead forget the dead in Haides, even there I will remember my beloved comrade" (Iliad 22.389ff).

In Odyssey, the dead generally do not remember anything until Odysseus permits them to approach and drink from the blood of the sacrifice he has made. Then they recall everything and are sorrowful. In the long line of visitors Odysseus greets while in the Underworld, some (like his mother) drink the blood and remember, while others just seem to remember on their own (like Achilles). But, if not properly buried and put to rest, the dead are unable to properly "settle" in the underworld and must come back to haunt the living world in search of release. They remember everything:

"There appeared to [Achilles] the ghost of unhappy Patroklos, identical in appearance and height, those lovely eyes and that voice, and it wore the same clothing as Patroklos had. The ghost came and stood over his head and spoke a word to him: ‘You sleep, Achilles, and you have forgotten me, but you were not careless of me when I lived—only in death. Bury me as quickly as possible, and allow me to pass through the gates of Hades. The souls, eidôla of dead men, hold me at a distance, and will not let me cross the river and mingle among them, but I wander as I am by the wide-gated house of Hades. And I call upon you in sorrow: give me your hand, for no longer will I come back from death to you, once you give me my funeral pyre.’" (Iliad 23.63).

And then Achilles reacts:

Achilles with his own arms reached for him, but could not grasp him, but the ghost went below with an empty wail, like mist, and Achilles jumped up, staring, and drove his hands together, and spoke with sorrowful words: 'Amazing! Even in the house of Hades something of a man remains, a soul and an image, but there is no real heart of life... (Iliad 23.99ff)

Christian conceptions of Hell derive mostly from the Roman era, especially from Vergil's descriptions of the underworld, which of course were very much indebted to the Homeric descriptions. In his Aeneid book 6, he describes many aspects of the underworld which are slightly modified from the Homeric version, including a fantastic description of Tartarus:

Aeneas looks back suddenly and beneath a dread cliff (sub rūpe sinīstrā) he sees broad walls surrounded by triple ramparts, which a swift river circles with roaring flames (quae rapidus flammīs ambit torrentibus amnis)—the river Phlegethon of Tartarus, tumbling about roaring rocks. There is a massive gate, and columns of solid adamant, so that no power of men can harm them with violence, and not even the power of the sky-dwelling gods (caelicolae); an Iron Tower stands there, and sitting upon its heights Tisiphone [the Fury], clothed in a bloody mantle (pallā succincta cruentā), guards the entrance night and day without rest. From here are heard groans and the savage lashes of the whip ring out, then the shriek of iron and of dragging chains..." (Aeneid 6.548ff).