There are seven figures on the P/V and, as Michael Turner pointed out, only one of them is unmistakable: Eros. As to the identities of the other six, opinions vary...
Scholars have linked them to Alexander Severus, "The Illiad", the Three Muses (though one cameo is clearly male!), the three goddesses judged by Paris (ditto!), and Leda with the swan (it's a snake!). One Christian reading holds that "scene one" shows Adam, Eve, an angel and God. It fails to account for "scene two".
A theory still popular today is that the animal is a sea serpent, indicating the figures are the parents of Achilles, the maritime deities Peleus and Thetis.
Potty potter Josiah Wedgewood claimed the first scene depicted a deceased person entering Elysium, a symbol of immortal life, and Hades. His explanation of the second scene, which began with another dead soul, was equally dubious.
In 1950, a "propagandist" interpretation was put forward, reasoning that since the vase was created during the time of Augustus it must relate to him. Therefore, the six contentious figures are: Apollo (the supposed divine father of Augustus), Atea (his real mother), Kronos, Apollo again, Atea again and Aphrodite.
Yet another flawed analysis states that, from the left in scene one, the characters are Antony, Cleopatra (plus asp), Anton (an invented son of Heracles), Antony's brother-in-law Octavian, his mourning wife Octavia and her patron deity.
[It occurs to me that I might be misremembering and/or misrepresenting the above arguments, but trust me when I say they were unconvincing even in their entirety. You might also quibble with my nomenclature - I've tried to stick to Greek titles, where Mr Turner often gave both the Greek and Roman versions, eg. Eros/Cupid.]
"Imagery has a purpose!" insisted the learned speaker. "And the vase was almost certainly placed within the grave for a reason."
He proceeded to relate the story of Adonis, so handsome that he was fought over by goddess of love Aphrodite and queen of the underworld Persephone. In the end, Zeus decreed the women could each have the youth for four months of the year, while Adonis would get the remaining four to himself.
M/T explained how the fate of Adonis represents a myth of death and rebirth - he is a god who dies and is reborn annually.
So, in the first scene, we have him entering the underworld (his cloak [not visible in the Wikepedia pic] is caught on a column). Leading him on is Persephone (the snake is a "chthonic" marker), and on the right is her husband, Hades.
In the second, we have Adonis again. The edifice is now in ruins, symbolically pulled down by his cloak. Persephone remains the central figure, but her torch and therefore the darkness of the underworld have gone out. The character on the right is Aphrodite. Adonis is gazing directly at her and ignoring the death goddess. It is clear where his true affections lie.
"The importance of Adonis and Aphrodite has been misunderstood."