What quotes in the play prove Hamlet was depressed?

An additional quote that shows Hamlet to be depressed comes at the beginning of his monologue in act 2, scene 2: "I have of late—but / wherefore I know not—lost all my mirth." For some reason, Hamlet has recently lost all sense of fun and joy. He thinks of the world as a sterile, empty place. For good measure, he has no time or affection for the human race either.

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By the time Hamlet gives his speech in Act 2, Scene 2, it's clear that he's completely disillusioned with the world. And this is clearly a by-product of his depression. As he readily admits to Rosencrantz and Guildenstern, he's recently lost all sense of fun:

I have of late—but
wherefore I know not—lost all my mirth...(II, ii, 250-51).

As Hamlet is so obviously depressed, it's not surprising that he should have such a jaundiced view of the world. Even the joys of nature appall him. Far from the sky being a majestic roof decorated with golden sunlight, it appears to the miserable Hamlet as nothing more than a "sterile promontory" whose very air is contaminated by a "foul and pestilent congregation of vapors". The very air that Hamlet breathes makes him feel sick to the stomach.

And Hamlet doesn't have much time for people, either. Though man is supposed to be a noble creature, unlimited in capacity for thought, angelic in action, godlike in understanding, he appears to Hamlet as nothing more than a "quintessence of dust."

There's clearly an element of self-loathing here. Hamlet hates himself for not moving quickly to avenge his father's death. So his misanthropic outburst about humankind can be seen as an attack upon himself. He recognizes that he, too, is nothing more than a "quintessence of dust."

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When Hamlet's mother confronts him about his continued grief over his father's death, she asks why his grief "seems [...] so particular" with him when he agrees that it is common for a son to lose his father. He replies, saying that it does not "seem" particular but, rather it is particular with him. He references his mourning clothes, his tears, his dejection:

Together with all forms, moods, shapes of grief,
That can denote [him] truly. These indeed "seem,"
For they are actions that a man might play;
But [he has] that within which passes show,
These but the trappings and the suits of woe. (1.2.85–89)

Hamlet is incredibly distraught not only by the loss of his father, but also by the very quick remarriage of his mother to her own brother-in-law who is, according to canon law, her actual brother. This makes the relationship not only inappropriate but incestuous. Hamlet says that the grief he has within is much more than show, much more than the mere trappings and appearance of grief. This seems to support the idea that he is, actually, depressed.

Further, after his mother and stepfather/uncle leave the scene, Hamlet says to himself:

O, that this too, too sullied flesh would melt,
Thaw, and resolve itself into a dew,
Or that the Everlasting had not fixed
His canon 'gainst self-slaughter! O God, God,
How weary, stale, flat, and unprofitable
Seem to me all the uses of this world. (1.2.133–138)

He actually wishes that his skin and bones would melt away so that he could die or, alternately, that God had not declared suicide is a sin. Hamlet feels that the world he lives in is worthless and without value, that it has nothing to offer to make life feel fulfilling.

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Hamlet learns from his father's ghost that his uncle, Claudius, murdered his father. Claudius is now married to Hamlet's mother. Hamlet has to decide if the ghost is really telling the truth. If so, Hamlet is honor-bound to avenge his father's death by killing Claudius. This situation depresses him.

Three indications of depression are seeing the world as an unpleasant place, showing sadness, and having thoughts of suicide. Hamlet shows all three.

Hamlet expresses his conviction that the world is rotten and corrupt in Act I when he says,

"How weary, stale, flat and unprofitable, Seem to me all the uses of this world! ...things rank and gross in nature / Possess it [the world]."  

In Act II, he says "the air ... appears no other thing to me than a foul and pestilent congregation of vapours," a powerful way of saying "the world stinks." He doesn't even want to breathe the same air as the corrupt courtiers Rosencrantz and Guildenstern. 

When his mother, Gertrude, notes that he looks sad, he responds that his looks don't begin to express the grief he feels inside. He calls "the fruitful river in the eye" [his tears] and his "dejected" face, "but the trappings and the suits of woe." In other words, while he is crying and he is sad, anyone can fake these feelings with false tears and sad face, but he, Hamlet, really feels sad within. 

Finally, Hamlet also expresses thoughts of suicide. In Act I, he says:

Oh, that this too too solid flesh would melt
Thaw and resolve itself into a dew!
Or that the Everlasting had not fix’d
His canon ‘gainst self-slaughter! O God! God!

In other words, he wishes he would melt to nothing or that God allowed people to commit suicide. 

In Act III, Hamlet again wishes he could die. As he puts it:

To die, to sleep—
No more—and by a sleep to say we end
The heartache and the thousand natural shocks
That flesh is heir to—’tis a consummation
Devoutly to be wished!
To Hamlet, the world is dull, flat and corrupt. Hamlet is very sad and he is flirting with suicide: we can safely say he is depressed.
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