How does Hamlet cause the deaths of the ones he loved?
To answer this question, we need first to consider just who it was that Hamlet loved. Clearly he loved his father, but he was in no way responsible for his death. He loved his mother as well, though he had, in a sense, fallen out with her after his father's murder and her remarriage. Accepting that he loved her, however, it is still difficult to say that he caused her death. Rather it was the machinations of Claudius that led to her death, which was in any case the result of a long chain of events set in motion by her marriage. He deeply cared for, if not loved, Horatio, but Horatio survives the carnage at the end of the play, and indeed Hamlet beseeches him not to take his own life. So essentially we are left with Ophelia.
Hamlet's love for Ophelia at the beginning of the play seems to be sincere, but Polonius orders her not to respond to it. This creates a tension between the two characters that explodes in Act III, Scene 1 when Hamlet, denying that he has loved her, shouts, "Get thee to a nunnery!" Ultimately, Hamlet's inadvertent murder of Polonius, combined, perhaps, with the stress of her very difficult relationship with the title character, causes Ophelia to lapse into madness, as a gentleman of the court describes in Act IV, Scene 5:
She speaks much of her father; says she hears(5)
There's tricks i' the world, and hems, and beats her
Spurns enviously at straws; speaks things in doubt,
That carry but half sense.
Later, Ophelia drowns outside the castle, having thrown herself, or perhaps fallen, into a stream. There is little doubt that Polonius's death played a role in causing her madness, and ultimately death, but of course Hamlet did not know what he was doing when he stabbed his sword into the arras, and really her death was down to her own father's scheming more than anything Hamlet did. So it is difficult to say, really, that Hamlet caused the deaths of anyone he loved in the play.