Both the Zeffirelli and the Olivier version of Hamlet made some changes to the play to make suggestions about Hamlet's relationship to different people.  Zeffirelli changed the order of some scenes and removed and shortened several speeches while Olivier changed less of the original,although he did change some speeches.

The two directors treat Hamlet's relationship to Claudius slightly differently.  In Zeffirelli's version,
Hamlet suspects Claudius' deception early.  He expects the worst of Claudius even before meeting the ghost.  This suspicion is exaggerated by the changed order of some of the lines.  Hamlet's early relationship to Claudius in the Olivier production seems to be one of disrespect.  When Claudius first speaks to Hamlet, he ignores him and does not answer until Gertrude speaks.  Later in the play, Hamlet's refusal to address Claudius as father seems to be done to irritate Claudius, and it does anger him.

Zeffirelli presents a much more affectionate relationship between Claudius and Gertrude.  He has them kiss relatively early in the play, in front of Hamlet.  This kiss prompts Hamlet to comment on the frailty of women.  In Olivier's version they show very little physical affection for each other.  The most physical attention between the couple appears at the beginning when Gertrude holds Claudius' hand while he announces their marriage.  By the end of the play, Gertrude's portrayal is very similar in both productions.   She becomes distrustful of Claudius during Hamlet's duel with Laertes.  In Olivier's version, she notices the cup early, and the audience wonders if she suspects poison before she drinks it.  

The Ghost is portrayed similarly in both productions.  Hamlet is drawn to him despite his fear. Zeffirelli removes the ghost's armor, which causes him to be much more personal and helps Hamlet, and the audience, see him as a suffering spirit.  The armor on Olivier's ghost hides some of his features and makes him appear more frightening.  In the Zeffirelli version,the ghost reaches out to Hamlet at he very end of his speech, but disappears before he would be close enough to touch him.  In Olivier's version, Hamlet reaches out toward the ghost when he is leaving.  Both productions portray Hamlet's relationship with the ghost as conflicted.  He loved his father, but he fears his ghost.

Both plays demonstrate conflict in Hamlet's relationship to women.  He struggles with his relationship to his mother and Ophelia.  Because he believes his mother remarried too early, he appears to distrust all women. Yet, in both productions he is often very affectionate toward Gertrude.  Zeffirelli emphasizes these conflicts by having Hamlet move away from Gertrude's affections while he mourns his father, but when Claudius leaves (another change from the original) he seeks comfort and his mother's embrace.  This suggests that his problem with Gertrude centers on his dislike of Claudius.  Olivier also shows some affection between Hamlet and Gertrude. Hamlet embraces Gertrude during some of their conversations.

Zeffirelli portrays Hamlet's relationship with Ophelia as very loving.  He shows a flashback of the incident Ophelia describes to Polonious, and she appears more confused than frightened.  Zeffirelli's changes to the nunnery speech really emphasize his love for Ophelia.  The lines asking her to go to a nunnery are not spoken until late in the play, when only Ophelia will hear them.  He speaks them as a warning that people are bad and she should not trust men.  This is clearly a  warning because he has recently decided not to trust Claudius.  The earlier part of the speech appears to be an exaggerated act.  Hamlet knows he's being watched so he speaks harshly to Ophelia so that her father will hear. Olivier also suggests that much of the harshness in Hamlet's speech is an act.  He is not cruel toward Ophelia until after he asks where her father is.  Her hesitation could cause him to be suspicious of her and by the end of his speech, Hamlet seems to know that someone is watching.  Both productions' portrayal of these lines as a performance of Hamlet's suggest that the feelings behind them are not true and allow the audience to believe Hamlet truly loves Ophelia.