Petrarch's Sonnet 333 is a fairly accurate description of Hamlet's feelings towards Ophelia when he learns of her death in Act V, Scene One. But there are differences when considering the entire plot of the play.
Prior to this scene, Hamlet has given Ophelia mixed signals. He had professed his love to her. Upon hearing of this, Polonius and Laertes instruct Ophelia to be cautious with Hamlet and Polonius even goes so far as to tell her not to talk to Hamlet.
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. (I.iv.132-34)
At the same time, Hamlet is feigning madness and he is distrustful of women and marriage in general because of his mother's hasty marriage to Claudius: "O most wicked speed, to post / With such dexterity to incestuous sheets!" (I.ii.156-57)
So, the break between Hamlet and Ophelia is a combination of these three converging events: Hamlet's false madness, Ophelia's instruction from Laertes and Polonius, and Hamlet's disenchantment with women and marriage which results in his poor treatment of Ophelia.
However, when Hamlet does finally learn of Ophelia's death, he is overcome, he remembers his love for her, and he is even jealous, claiming he loved her more than Laertes (her brother) ever could:
I loved Ophelia. Forty thousand brothers
Could not, with all their quantity of love,
Make up my sum. (V.i.254-56)
In Petrarch's Sonnet 333, the speaker, like Hamlet, is also at his beloved's (Laura's) grave. The difference is that, in the sonnet, this is some time after she's died because the grass is overgrown; Hamlet proclaims his love for Ophelia as she's being buried.
Petrarch's second stanza echoes Hamlet's "to be or not to be" speech:
Tell her, I'm sick of living; that I'm blown
By winds of grief from the course I ought to steer,
That praise of her is all my purpose here
And all my business; that of her alone
In the context of Ophelia's funeral scene, Hamlet praises his love for her; and like the speaker in the sonnet, this is Hamlet's only purpose at that moment. But in the wider context of the entire play, Hamlet's "business" is "steered" by his plan for revenge. In the sonnet, Petrarch's purpose is only to proclaim his love for Laura. Hamlet's primary purpose is to seek revenge. In the process of that revenge, he questions whether to live or die, not in regards to Ophelia, but in regards to killing Claudius and completing his revenge.
In the fourth stanza, the speaker wonders if Laura, in spirit form, would "deign to stand at my bedside" at his death. In this context, the speaker wonders if such an angelic creature such as Laura, in life and in death, would deign (condescend, do something that is beneath her, or humble herself to such a position) to be at his side at his death. Hamlet makes a somewhat similar declaration when he is arguing with Laertes at Ophelia's grave:
Be buried quick with her, and so will I. (V.i.264)
The significant similarity between Sonnet 333 and Hamlet's feelings toward Ophelia is most evident in Ophelia's funeral scene. But since Hamlet was preoccupied with avenging his father and other political implications, only part of his attention was devoted to Ophelia. His dying words are not about Ophelia; instead, he supports Fortinbras as the next leader. So, it seems that Hamlet was not as solely focused on his beloved as Petrarch's speaker was.