What did the Greeks believe about the person's soul after they died?

After death, the soul traveled to the Underworld and was ferried across the river Acheron by Charon (all dead people needed a coin with which to pay the ferryman). Once across the river, they went to one of three places, dependent on their character in life:
The Asphodel Meadows, where ordinary people mostly ended up
Tartarus, for those who had been truly terrible in life
Elysium/the Elysian Fields, for heroes
Souls are described as “shades,” literally like shadows of their living counterparts, and like shadows they lack any strength or ability to influence the world of the living; they’re insubstantial and aimless. They’re pretty much exactly as they were in life at the moment of their death—so people who died in battle have dripping wounds, for example—but nobody really feels anything or thinks anything, they’re just stuck. They are very two-dimensional. The Underworld is effectively “limbo” in the Christian sense; nothing happens except to the very good or the very bad. The only exceptional thing about the souls of the dead is that they stand outside of Time and consequently can foretell the future, which is why Odysseus goes to the Underworld (likewise Aeneas). Consulting the dead for advice is pretty much the only influence they can have on Events. Otherwise they are static, listless, aimless, and experience neither suffering nor any kind of heavenly reward.
Specifically relevant to Antigone is the fact that there was a general belief that bodies must be disposed of appropriately, because leaving them unburied (or un-cremated) was a serious taboo and would result in the soul of the dead person not being allowed to cross the Acheron (Charon would refuse them passage so they’d be stuck in a permanent waiting area).