How can you analyze the physical settings of both The Great Gatsby and Hamlet as metaphorical backdrops for the psychological states of three of the characters of the chosen settings?

To begin, The Great Gatsby is set in New York state, in a geographical area that includes West Egg, East Egg, and the valley of ashes. West Egg is where Gatsby lives: it is the "less fashionable" of the two eggs, according to Nick Carraway, the narrator. Nick later describes West Egg, in Daisy's eyes, as an

unprecedented 'place' that Broadway had begotten upon a Long Island fishing village—[and she was] appalled by its raw vigor that chafed under the old euphemisms […].

This setting serves as a metaphorical backdrop for Jay Gatsby's psychological state. The fact that Daisy considers "Broadway" to have "begotten" West Egg matches perfectly with the owl-eyed man's comparison of Gatsby to David Belasco, a popular theatrical producer, director, and playwright. Gatsby is adept at putting on a show, at creating a realistic spectacle, at pretending to be something he is not, just as West Egg and its inhabitants try to pretend that it is East Egg: something it is not, as Daisy is well aware. The inhabitants of both Eggs are hedonistic, but those who live in West Egg—the "nouveau riche" or people with new money—seem to "chafe" against the rules about what one says or does in good society (whereas, for people like Daisy, conforming to these rules is old hat). Gatsby doesn't understand certain old money social cues or standards, just as he doesn't understand that he can never raise himself to Daisy's social class no matter how good he is at pretending. West Egg is the perfect setting for both his desire to raise himself up and his inevitable failure to do so.
East Egg, on the other hand, is filled with "white palaces" that "glittered along the water." The old money, upper-class live there, and this place serves as a metaphorical backdrop for Daisy Buchanan's psychological state. She is untouchable, privileged to a degree that most of us cannot begin to imagine. When Gatsby describes her, late in the novel, he says that "Her voice is full of money." Nick agrees, saying,

that was the inexhaustible charm that rose and fell in it, the jingle of it, the cymbals' song of it . . . . High in a white palace the king’s daughter, the golden girl . . . .

Daisy is just like East Egg, glittering and effortlessly rich: untouchable and gorgeous. She is so mentally in-line with this location that she cannot bring herself to sacrifice it, even for Gatsby, the man she seems to love the most.
The valley of ashes, finally, serves as the perfect metaphor for George Wilson. Nick describes it as

a fantastic farm where ashes grow like wheat into ridges and hills and grotesque gardens; where ashes take the forms of houses and chimneys and rising smoke and, finally, with a transcendent effort, of men who move dimly and already crumbling through the powdery air.

The valley of ashes is often thought to be representative of the effects of unbridled industry and the unethical treatment of the lower classes by the upper, and we certainly see this echoed in the treatment of George Wilson by Tom Buchanan. Tom has, evidently, been promising to sell George a car that George can then fix up and turn around to make a profit, but Tom has not exactly been forthcoming, and he seems to dangle it like a carrot in front of George. On top of this, Tom is having an affair with George’s wife, Myrtle. In this sense, then, Tom takes from George both professionally and personally; he doesn’t seem to see George as a person at all until after Myrtle's death. Further, Nick describes George as "a blond, spiritless man, anemic, and faintly handsome." George even sounds dusty and ashen, especially compared to his "smouldering" wife and her "immediately perceptible vitality." In many ways, she is described like something that burns, while her husband is described like something that’s been burned: just like the valley itself.
Rather than examining a few different locales, the one setting of Hamlet—Elsinore, Denmark—will actually serve as a metaphorical backdrop for several characters. Marcellus, one of the watchmen who sees the ghost of old King Hamlet, famously says, "Something is rotten in the state of Denmark" (1.4.94). Certainly, the first character to whom this applies is Hamlet, the young prince. He describes how, since his father's death and his mother's hasty remarriage to Hamlet's uncle (his father's own brother), the world has seemed to him "an unweeded garden / That grows to seed. Things rank and gross in nature / Possess it merely" (1.2.135-137). For him, innocence has been lost. He compares Denmark to a garden that has been left to rot, that has become overgrown with weeds and dead plants—much like the Garden of Eden after Adam and Eve were kicked out. In his mind, the paradise that existed before his father's death has been blighted, and neither its beauty nor Denmark's innocence can return, with his (biblically) incestuous mother and uncle on the throne. Hamlet's own mental health has been materially damaged as well, and he now contemplates suicide as a solution to end his own misery.
When Claudius, Hamlet's uncle, killed his own brother—in a garden—in order to steal his brother's wife and crown, Claudius sullied his own mind, just as he has the throne of Denmark. One night, while alone, he soliloquizes, "Oh, my offence is rank. It smells to heaven. / It hath the primal eldest curse upon 't, / A brother’s murder" (3.3.37-39). It cannot be coincidence that he uses Hamlet’s word ("rank") to describe what he's done. He does feel guilty, but he wants to retain those things he gained when he killed his brother, and so he finds it difficult to pray for forgiveness. He has dragged the court into corruption with him, corruption that must be totally cut out of Denmark in order for the country to prosper and unite again. His actions eventually lead to his countrymen calling for Laertes, Polonius's son, to take the throne away from him. In the midst of political discord and civil unrest, problems Claudius has created, he is conflicted within himself as well, and so the physical setting mirrors his psychological state too.
Finally, once Hamlet brings his mother, Gertrude, around to his way of thinking about Claudius, her own mind comes to reflect the unsettled and fallen paradise that is Denmark. She bids him,

O Hamlet, speak no more! / Thou turn’st mine eyes into my very soul, / And there I see such black and grained spots / As will not leave their tinct. (3.4.89-92)

In other words, then, when Gertrude looks into her own soul and conscience, she now sees that they are tainted by her sinful and selfish behavior. She feels so terrible that Hamlet’s father’s ghost asks Hamlet to "step between her and her fighting soul" (3.4.114). Thus, she clearly feels the same sense of unrest and discord within, as Denmark will soon possess without. For her, too, paradise and innocence are lost and can never be regained.