how does the speech about priam mirror parts of the main plot in hamlet

Hamlet may possibly be seeing himself in the character of Pyrrhus, who is avenging his father's death. However, all the other circumstances are quite different. Hamlet is far from the bloody warrior described in this speech, and Priam is no Claudius, being characterized as an ancient, helpless man, while Claudius is in his vigorous prime.
When the Players first appear, Hamlet requests the First Player recite the "Priam" speech, which Hamlet recalls from seeing it at one point, and this leads to the famous soliloquy that begins: "Now I am alone. / O, what a rogue and peasant slave am I!" in which Hamlet mocks his own inability to avenge his father's death, while the Player can "force his soul so to his own conceit" that he weeps for someone dead centuries ago.
Of course, later, the play-within-a-play, called by Hamlet "The Mousetrap," is deliberately constructed to mirror the circumstances of Hamlet's father's murder by his uncle (and the faithlessness Hamlet sees in his mother), so perhaps that is actually the section of the play to which you refer.