How does the play-within-the-play reflect the issue bothering Hamlet, and which are the lines Hamlet has inserted?
The play, "The Mousetrap", reflects the rage Hamlet feels toward Gertrude for marrying Cladius. He feels betrayed by her actions.
The Player King tells his Queen that she might find another husband after his death. The Queen responds, "Oh, confound the rest! Such love must needs be treason in my breast. In second husband let me be accursed! None wed the second but who killed the first." The Queen is saying that, remarrying would be treason to her heart. She says that when a woman takes a second husband, it's because she's killed the first.
The Queen continues: " A second time I kill my husband dead when second husband kisses me in bed." The Queen says anytime she kissed her second husband in bed, she'd kill the first one all over again.
Later, the Queen continues: "Nor earth to me give food, nor heaven light. Sport and repose lock from me day and night. To desperation turn my trust and hope. An anchor's cheer in prison by my scope. Each opposite that blanks the face of joy meet what I would have well and it destroy. Both here and hence pursue me lasting strife if, once a widow, ever I be a wife!"
The Queen is saying may the earth refuse me food, and the heavens go dark, may I have no rest day and night, may my trust and hope turn to despair, may the gloom of prison overtake me, and may my every joy be turned to sorrow. May I know no peace either in this life or the next one, if I become a wife again, after I am a widow.
The play-within-a-play with a king and a queen mirrors the reality of King Claudius and Queen Gertrude's actions. In response to Claudius, Hamlet explains that the play is based upon a murder in Vienna: Gonzago is the duke, Baptista his wife. The player-king tells his wife that he is going to die; the queen refuses to believe. Refuting what she says, the player-king tells her these ideas will die when he dies, and she will wed another.
When the character named Lucianus enters, he mentions the ingredients of his poison. Then, he pours this mixture into the ear of the player-king, just as Claudius did to Hamlet's father. Lucianus' lines represent the lines Hamlet inserted:
Thoughts black, hands apt, drugs fit, and time agreeing,
... On wholesome life usurps immediately (III.ii 232-238)
As art too closely imitates life, Claudius is greatly disturbed and demands that the play be ended: "Give o'er the play....Give me some light. Away!" (III.ii 245-246). The King exits.