How is the literary theory of feminism portrayed in Hamlet?

Feminist literary theory (or feminist literary criticism) revolves around the idea that literature reflects and shapes ideas about men and women and their roles in society. This form of criticism engages with a range of key ideas and questions—and provides an interesting lens through which to consider Shakespeare's Hamlet. Let's go over a few of those ideas and how they apply to the text, with examples.
How are female characters portrayed in the text? To what extent can the reader identify with female characters?
Let's start with Ophelia, Hamlet's love interest, who eventually goes mad. She's not exactly a powerful woman—she's young and innocent, and people talk about her beauty more than any other detail. She's obedient to her father. She doesn't have much to do. Early in the text, she's relatable enough, but this characterization of a young woman seems antiquated. Where's her agency and ambition?
Later, after her father dies and Ophelia loses her sanity, she delivers a few memorable speeches about how terrible young men are—how they're after one thing, and they're not faithful to women, either.

CLAUDIUS
Pretty Ophelia—
OPHELIA
Indeed, without an oath I’ll make an end on ’t:
(sings)

By Gis and by Saint Charity,
Alack, and fie, for shame!

Young men will do ’t, if they come to ’t.

By Cock, they are to blame.

Quoth she, “Before you tumbled me,
You promised me to wed.”

From a feminist critical perspective, this is pretty interesting. Everyone says she's acting crazy, and that she's lost her mind (and yes, she has) partly because she is daring to say something critical about men instead of just obeying their orders.

How are relationships between men and women described? What does it suggest about power, sexuality, and gender?
Let's look at this passage. Note that I am condensing this passage for clarity, indicated by the ellipsis:

CLAUDIUS
Sweet Gertrude, leave us too,
For we have closely sent for Hamlet hither,
That he, as ’twere by accident, may here
Affront Ophelia.

GERTRUDE


I shall obey you.
And for your part, Ophelia, I do wish
That your good beauties be the happy cause
Of Hamlet’s wildness.



There's a lot going on here. Claudius asks Gertrude to leave, indicating that she's not needed or wanted, and she "obeys." (Note that she is THE QUEEN, by the way.) But not before delivering this line to Ophelia about how she hopes that the young woman's beauty will somehow calm down Hamlet's crazy behavior.

In this passage, Gertrude is docile, and she reinforces the notion that women are to be seen, not heard. (And that one of the best things a woman can be is beautiful.) Not exactly a rousing feminist statement.

Does the piece of literature reinforce the oppression of women, whether politically, socially, economically, or socially?
The depictions of both Ophelia and Gertrude are problematic. We've already talked about Ophelia, above, so let's focus on the fact that we, as the readers, only really see the character of Gertude through the perspective of Hamlet, her son. And Hamlet is very troubled by the notion that his mother is a sexual being. He attacks her for remarrying (Claudius is now her husband) and even of killing her husband, Hamlet's father. But his obsession with her sexual life is notable, too:

GERTRUDE
O Hamlet, speak no more!
Thou turn’st mine eyes into my very soul,
And there I see such black and grainèd spots
As will not leave their tinct.

HAMLET



Nay, but to live
In the rank sweat of an enseamèd bed,
Stewed in corruption, honeying and making love
Over the nasty sty—


This pretty much speaks for itself: Hamlet wants to force his mother into chastity. He doesn't look so good through a feminist critical lens, does he?
http://artsites.ucsc.edu/faculty/bierman/elsinore/women/womenPortraits.html