The Greek Underworld
Hades and Persephone in the underworld
Greek people did not like to talk about what happened to you after you died, and so we don't know as much about what they thought as we might like. Probably they did not think about it as much as later people like the Christians and the Buddhists did, or at least not in the same ways.
Most Greeks believed that everybody had a spirit, which lived on after your body died, and they thought of this spirit as being rather like our ghosts, sort of transparent and floaty, but looking like the living person otherwise.
They thought that you had to do certain ceremonies when somebody died, in order to let their spirit go to the land of the dead. If you did not do these ceremonies, their spirit would continue to hang around the land of the living, haunting you and making a nuisance of itself. The most important thing was that dead people had to be buried. (see the story of Antigone, for instance).
Persephone supervises the punishment of Sisyphos
After your body had been buried, so you were under the ground, you could cross an underground river, the river Styx (pronounced STICKS), to get to the land of the dead. Often dead people were buried with a small coin or two to pay the ferryman, Charon (KA-ron) who took you across the river Styx.
Then when you got to the land of the dead, it was basically like any place underground: dark, damp, and chilly, with nothing much to do, and lots of ghost-spirits floating around, bored and depressed, sounding like thousands of bats. You just stayed there forever. There was no promise of a better place for good people, or a worse place for bad people. (Though there were exceptions like Tantalus and Sisyphus, who had the hubris to think he could trick Zeus.) Hades was king there, and sometimes Persephone was the queen.
When Odysseus visited the land of the dead and saw the spirit of Achilles there, he asked him what it was like, being dead, and Achilles said that he would rather be a landless field hand, and alive, than be the king of the dead.
This description is about the same as what other ancient people thought around the same time. The Zoroastrians and the Jews had similar ideas. Gradually people got more interested in the afterlife, and by the time of Jesus everyone - not just Christians - was beginning to think about a different afterlife for good and bad people.
Learn by doing: draw a picture of the Greek underworld
More about Hades
Persephone and the Pomegranate: A Myth from Greece, by Kris Waldherr (1993). . Not cheap, but beautifully illustrated.
D'aulaire's Book of Greek Myths, by Edgar and Ingri D'Aulaire.
Greek Religion, by Walter Burkert (reprinted 1987). By a leading expert. He has sections on each of the Greek gods, and discusses their deeper meanings, and their function in Greek society.
Aspects of Death in Early Greek Art and Poetry, by Emily Vermeule (1979). She's an expert on early Greece, and this book goes into detail about what the Greeks thought happened to people after they died. For serious readers.