she can have a relationship with the prince. He commands her to have no further relations with him:

I would not, in plain terms, from this time forth,Have you so slander any moment leisure,As to give words or talk with the Lord Hamlet.Look to't, I charge you: come your ways.

Polonius also believes Hamlet is going mad and seems to be concerned about his seeming obsession with Ophelia. When Ophelia informs him of Hamlet's verbal declarations of love, the gifts he gave her, and the affectionate letters he wrote her, Polonius insists Ophelia return them.

In Act II, Scene 2, Polonius reports Hamlet's strange behavior towards Ophelia to Claudius and Gertrude, telling them he disciplined his daughter by telling her, 

'Lord Hamlet is a prince, out of thy star;This must not be:' and then I precepts gave her,That she should lock herself from his resort.

When Polonius later speaks to Hamlet, he takes particular note of his constant references to Ophelia and assumes Hamlet is obsessed with her and mentally unstable. 

In Act III, Scene 1, Ophelia confronts Hamlet and tells him that she wishes to return all the "remembrances" he had given her. Hamlet denies that he had given her anything. During their conversation, the prince gets quite upset and insults Ophelia, telling her: "get thee to a nunnery," for she should not breed sinners. He accuses her of lying. Ophelia is obviously unnerved about his ranting and calls upon the powers of heaven to restore him. Hamlet storms off.

Polonius and Claudius have been eavesdropping on the conversation and then discuss Hamlet's behavior. They agree that all is not well with the young prince. Claudius undertakes to send him to England, while Polonius advises that the king should ask Gertrude to consult with her son. If Gertrude cannot make any headway, Polonius says Hamlet should be sent to England. Claudius ends the scene by stating,

It shall be so:Madness in great ones must not unwatch'd go.

Polonius reads Claudius and Gertrude a letter Hamlet sent to Ophelia. The gist of the short letter is to reiterate how much he loves her. Hamlet uses hyperbole, or exaggeration, as the letter opens, saying she should doubt such certainties as that the stars are made of fire and the sun moves before she doubts his love for her. While he complains that he is not a good poet and that his verse can't convey the true depth of his feelings, he states plainly that he loves her:

I love thee best. Oh, most best, believe it.

On the whole, Hamlet is a very complex character. It is this complexity, more than anything else, that makes him so interesting. However, there is also a much simpler side to the young Danish prince, as can be seen in the love letter he writes to Ophelia.

Once we become aware of the contents of this letter, there can be no doubting the depths of feelings that Hamlet harbors for Ophelia. In the rhetorical fashion of the day, he says that Ophelia can doubt the movement of the sun and that the stars are fire, but she cannot doubt his love for her. It is clear from his love letter that Hamlet truly loves Ophelia and that there are no ulterior motives involved.

Ophelia's father, Polonius, seizes on the letter as evidence that Hamlet's strange behavior of late can be attributed to his love for Ophelia. Young people tend to do pretty strange things when they're in love, and so Polonius's theory is not all that implausible.

In any case, Polonius doesn't appreciate the heartfelt contents of the love letter; he looks upon them as so much soppy rhetoric that indicates Hamlet not to be in the right state of mind.

Why does Hamlet write a letter to Ophelia?

Hamlet has composed a letter to Ophelia for the very best purpose--to convey his love for her!  But wait, I thought Hamlet and Ophelia are estranged from each other during the entire play?  To solve this confusion, we must consider just who it is that brings the famous love letter to Claudius--the fawning Polonius, who will do anything to curry the king’s favor and prove his own worth.

To investigate, let’s back up a bit.  In act 1 scene 3 Polonius orders Ophelia not to speak with or see the prince any more, and she has no choice but to obey him. In act 2 scene 2 she tells Polonius, “as you did command / I did repel his letters and denied / His access to me.”  Therefore, she has not received any new letters from Hamlet since before he put “an antic disposition on.”  It's possible that Polonius, to add weight to his “Hamlet is crazy for my daughter” theory, takes an old love letter Hamlet had given her. When he tells the king and queen his theory in act 2 scene 2, he vaguely says that Ophelia has obediently given him the letter.  They merely assume it is a recent letter.

Although Polonius mocks the language of the letter as he reads it out loud to them, we can hear that Hamlet is quite gifted in rhetoric, something actually considered unfashionable among royalty in the late middle ages, which the play is set in.  Hamlet himself tells Horatio in act 5 scene 2 that he once felt somewhat embarrassed that he writes so well, and even tried to forget how. But the skill comes in handy when Hamlet needs to rewrite Claudius’ letter to the king of England, the “changeling” bringing about the death of Rosencrantz and Guildenstern instead of Hamlet himself. It is also quite handy for wooing maidens, it seems. 

So Hamlet wrote the letter to express his deep and honest love for Ophelia, something we would never know about if not for those words written by a free and untroubled prince, perhaps before his father ever died.  The most convincing lines might be these: “Doubt thou the stars are fire / Doubt that the sun doth move / ...But never doubt I love.”  He signed the letter beautifully with, “Thine evermore, most dear lady, whilst / this machine is to him, Hamlet.”  He vowed to love her as long as he lived.  More so than the question of his sanity, whether he loves Ophelia until his last breath may be the true secret that Hamlet takes with him to “the undiscovered country.”