What is the conceit(s) in Hamlet's first soliloquy in Act I, Scene 2?
Concisely, a conceit is an extended metaphor or unusual comparison. During Elizabethan times, conceits were common and normally flowery or representative of complicated logic.
In Act 1, Scene 2 of Hamlet, Hamlet first compares the world to a forgotten and overtaken garden, controlled by the grotesque of the earth. He claims that the world now only grows "things rank and gross in nature" (1.2.139). Not only does this conceit demonstrate Hamlet's suicidal thoughts which would enable him to leave behind so despicable a place as the earth as become to him, but it also hints at the relationship between his mother and uncle--Hamlet believes that what they are doing is "rank" and that it goes against nature.
A second conceit begins later in Line 143 when Hamlet compares his mother to a beast. He discusses her insatiable "appetite" and calls her a satyr.
Both conceits demonstrate that Hamlet views all the play's preceding action as unnatural--in essence, his father's death has turned the world upside down.
In poetry, a conceit is an extended metaphor. Certainly in Hamlet's first soliloquy, there are some metaphors, but, as a whole, the speech does not have a single, continuous extended metaphor.
If this helps, here are the metaphors:
First Hamlet compares himself to water or ice:
O, that this too too sullied flesh would melt,
Thaw and resolve itself into a dew
Then he compares the world to a neglected garden:
How weary, stale, flat and unprofitable
Seem to me all the uses of this world!
Fie on't! ah, fie! 'tis an unweeded garden
That grows to seed; things rank and gross in nature
Possess it merely.
The rest of the soliloquy is a rant against his mother, Queen Gertrude.
So, although there are metaphors, comparisons, there is no single, controlling metaphor or conceit.