Hamlet is a play not just about the young prince's troubled mind but a play about relationships. How can I analyze one of these relationships in the play through close readings of speeches or scenes and through my analysis of two secondary sources? I'm interested in the Gertrude/Claudius, Hamlet/Ophelia, and Horatio/Hamlet relationships. What are the hazards and benefits of this relationship and how does it connect with the broader themes of the play?

Queen Gertrude/King Claudius
While the relationship between Gertrude and Claudius appears to be simply one of lust and cupidity to Hamlet, there are, however, underlying complexities to this marriage that play into the theme of moral corruption.
Both share in guilt; Gertrude is guilty of lust, and Claudius of lust and fratricide. When King Hamlet's ghost talks with Hamlet, he accuses Gertrude of lust as he states that there was a "falling-off" of his "virtuous queen" to a "wretch whose natural gifts were poor" in comparison to his (1.5.47-51). Later, in Act III, Gertrude herself confesses to her lust when Hamlet confronts her about her relationship with Claudius:

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

Also in Act III, Claudius prays after having watched Hamlet's play, visibly shaken by the drama's closeness to reality. Hamlet overhears Claudius confess his guilt:

O, my offence is rank, it smells to heaven;It hath the primal eldest curse upon'tA brother's murder. Pray can I not,Though inclination be as sharp as willMy stronger guilt defeats my strong intent....(3.3.36-40)

Additional Sources:  
Bloom, Harold. Shakespeare: The Invention of the Human. New York:             Riverhead Books,1998.
Mackenzie, Henry. "Criticism on the Character and Tragedy of Hamlet." The Mirror 99 (1780): 18.

What an interesting question! For the purposes of this question, I will discuss Hamlet's relationship with his mother, Gertrude. This mother-son relationship connects to the themes of deceit and women's frailty in the play.
The benefits and hazards of Hamlet and Gertrude's relationship.
Hamlet has an interesting love-hate relationship with Gertrude. I think the main conflict between mother and son stems from Hamlet's analysis of Gertrude's seemingly hasty marriage to Claudius. Hamlet is suspicious about Gertrude's motives; he thinks that Gertrude has fostered an incestuous alliance with his father's likely murderer.
For her part, Gertrude loves her son, and she ostensibly marries Claudius to protect Hamlet's position in the kingdom. Gertrude feels that her marriage alliance will confer political and personal benefits to her son, benefits she has no power of securing otherwise. However, because of the sensual underpinnings of Gertrude and Claudius' relationship, Hamlet feels that he can trust neither his mother nor his new step-father. Deceit is a political tool that engenders conflict between mother and son. However, deceit is possibly the only weapon left to Gertrude, who lived in an era when women had little personal power and agency. She must execute a tight balancing act between Claudius and Hamlet in order to protect her station in life. Hamlet's relationship with his mother definitely highlights the themes of deceit and women's frailty in the play.
From what we can see, Claudius has hidden his culpability in his brother's death from Gertrude. Gertrude is also less than forthcoming about Hamlet's actions to Claudius. In Act 3 Scene 4, Hamlet openly accuses his mother of betraying and insulting his father (King Hamlet), of lying "in the rank sweat of an enseamèd bed,/ Stewed in corruption, honeying and making love.." to a murderer (Claudius). He insinuates that Gertrude was a full participant in his father's death and is guilty of great sin. Gertrude doesn't deny the "black and grainèd spots" in her soul, but she stops short of confessing to being an accomplice in King Hamlet's murder.
Hamlet further torments his mother by accusing her of lewdness. He asserts that a woman of her age is past her sexual prime. It's a very sexist statement, of course. Again, the theme of female sexuality is highlighted in the tension between Hamlet and Gertrude. Hamlet cannot conceive of his mother having any sort of sensual preferences at her age. He tries to mitigate his inner torment about such a possibility by resorting to a chauvinistic explanation:

You cannot call it love, for at your age
The heyday in the blood is tame, it’s humble,
And waits upon the judgment (Act 3, Scene 4).

The relationship between Hamlet and Gertrude is essentially poisoned because Hamlet is unable to forgive his mother for marrying Claudius. He attributes her hasty remarriage to the "frailty" of women, a popular stereotype during Shakespeare's time. Hamlet is supported in his diatribe by his father's ghost. The ghost appears to steady his son's hand in avenging his death. Earlier in the play, King Hamlet's ghost contrasted the pure love he shared with Gertrude against the adulterous love Gertrude now shares with Claudius. This plays into the popular stereotype about higher-classed women desiring the more primitive, virile men of the lower classes. King Hamlet's ghost definitely views Gertrude's alliance with Claudius in the light of this stereotype. Hamlet does his father one better: he orders his mother to stop sleeping with Claudius altogether:
 

Good night—but go not to mine uncle’s bed.



Assume a virtue if you have it not.
That monster, custom, who all sense doth eat,
Of habits devil, is angel yet in this:
That to the use of actions fair and good
He likewise gives a frock or livery
That aptly is put on. Refrain tonight,
And that shall lend a kind of easiness
To the next abstinence, the next more easy (Act 3, Scene 4).


The events in Act 3, Scene 4 definitely illustrates the depth of the dysfunction in Hamlet's relationship with his mother. He tells Gertrude that he firmly believes Claudius had a hand in killing his father. However, neither openly discuss Gertrude's possible culpability in the whole affair. Gertrude promises to keep Hamlet's confidence and later, tries to deceive Claudius about Hamlet's guilt in killing Polonius. She tells Claudius that Hamlet "weeps for what is done (Act 4 Scene 1). Of course, this isn't true at all, but Gertrude is merely protecting Hamlet.
 
To recap, Hamlet's relationship with his mother definitely highlights the larger themes of deceit and women's frailty in the play. If you're looking for some good sources to back up this argument, I include two links below. Two of my sources are also listed here.
Sources:
1) Gertrude's Elusive Libido and Shakespeare's Unreliable Narrators by Richard Levin, Studies in English Literature, 1500-1900, Vol. 48, No. 2, Tudor and Stuart Drama (Spring, 2008), pp. 305-326
2) Hamlet and Gertrude or The Conscience of the Queen by Robert M. Smith, The Shakespeare Association Bulletin, Vol. 11, No. 2 (APRIL, 1936), pp. 84-92
 
http://elsinore.ucsc.edu/women/womenGertrude.html

Hamlet’s relationship with his mother, Queen Gertrude, is perhaps the most dynamic, controversial, and volatile in Shakespeare’s Hamlet.  His feelings toward his mother alternate between affection (some critics have interpreted this as both the filial kind and something much more Freudian), contempt, and disgust.
I'll discuss the Freudian reading of Hamlet, which is one of many different interpretations of Hamlet's relationship with Gertrude.
Hamlet’s need of a mother’s approval can be seen in act I, scene ii when he acquiesces, “I shall in all my best obey you, Madam.”  Yet in a deeply subconscious part of Hamlet’s psyche, those who read the text this way would argue that he desires Gertrude for himself.  Some would say this is an Oedipus reaction--the Freudian theory that all men secretly desire their mothers.  Others would suggest that Hamlet merely wants his mother to be the perfect, pure madonna.  He wants no man to have her, that she might be mother to him alone. The best scene in which to analyze these feelings of son for mother would be act III, scene iv.  A great source of analysis is the article “Doctor Freud’s Hamlet,” from the University of California-Santa Cruz website Art Sites.
As a prince and a man, Hamlet righteously feels contempt, even disgust towards the queen.  Key scenes for a deeper analysis here would be act III, scene ii (the play within the play) and again Hamlet’s deeply emotional outburst toward Gertrude in act III scene iv. An interesting source on this is Chikako D. Kumamoto’s article “Gertrude, Ophelia, Ghost: Hamlet’s Revenge and the Abject,” from the January 1 issue of the Journal of the Wooden O Symposium.
Considering his bewildering relationship with his mother, it is no wonder that Hamlet finds difficulty in acting out the requested revenge against Claudius. In his uncle he sees the “portraiture” of his own sinful views of Gertrude. Thus, Hamlet falls to self-deprecating soliloquies rather than revenge.
http://artsites.ucsc.edu/faculty/bierman/elsinore/freud/freudSolved.html

http://web.b.ebscohost.com/Legacy/Views/static/html/Error.htm?aspxerrorpath=/ehost/pdfviewer/pdfviewer

Shakespeare's characters and their relationships are a rich topic of discussion. It seems that you may choose any two characters for this essay and analyze their relationship. I would suggest that first you choose two of the more significant characters, and then do a close reading of scenes in which the two characters interact in order to get a true and complete picture of their relationship. Remember that Shakespeare's words often are sarcastic or ironic in tone, and meanings may be influenced much by the way the actors performing the play say the lines. In other words, you may need to read between the lines to determine what a character's intentions truly are.
As an example, you may choose to study father-son relationships in Hamlet. This would be interesting because you could choose either Hamlet and his dead father, or Hamlet and his uncle/new stepfather, Claudius.
An interesting secondary source to use in analyzing either of these father-son relationships would be "The Death of Hamnet and the Making of Hamlet," by Stephen Greenblatt, which is an essay positing that the death of Shakespeare's son Hamnet greatly influences his emotional and profound rewriting of what was before a very simple and somewhat worn out story of Danish royalty.
Another useful source could be the book Hamlet: A New Source, A New Reading by Ciriaco Morón Arroyo, which gives a very in-depth description of how Hamlet reacts to his uncle becoming his stepfather. Arroyo discusses the traditional archetypes of royalty and how Shakespeare rejects these stereotypes, instead showing how two royal brothers can be completely opposite from one another. This source would back up an analysis of either father-son relationship, showing how each paternal character affects Hamlet and how he compares them in his mind.
After you choose two characters to analyze, I would recommend using Google Scholar article search or a scholarly website such as jstor.org in order to find appropriate secondary sources. These resources will help you to find articles which discuss the scenes and characters you would like to analyze.