In Hamlet there are three key families which are the main focus in the play. The families are Hamlet's family, Polonius's family and Fortinbras's family. There are many similarities and differences between all three separate families. One similarity that all three families have is that at least one member of their family seem to have a hidden agenda. Hamlet wants revenge against Claudius and the reason Claudius did so was to become the new king of Denmark. Laertes wants revenge for his father Polonius who was killed by Hamlet. Lastly Fortinbras has a hidden agenda when he initially says that he wants to pass through Denmark to invade Poland but there are suspicions that he has a secret agenda to overthrow Denmark and reclaim the land that was lost when the old king of Denmark defeated his father and became king.
There is also spying in Hamlet between families and this shows that there is an absence of trust. An example would be when Polonius sends the spy Reynaldo to go and spy on Laertes as he suspects that his son is up to no good. He says "your bait of falsehood take this carp of truth", this is a metaphor and means simply use a subtle lie and catch the truth. Polonius is telling use the bait of lies to find out more information and more specifically the truth. This idea of telling a lie and finding out the truth can also be backed up when Polonius says "with windlasses and with assays of bias" which again can be interpreted as indirect attempts so by telling a lie so the truth comes out. "Windlasses" metaphorically means to decoy so for example decoy an animal by making an indirect approach, and so by Reynaldo is to tell a small lie as a decoy and as an indirect attempt to find out the truth.
This absence of Trust can also be seen in the family relationship of Hamlet and Claudius as Hamlet knows his uncle killed his brother to become the king and Claudius sees Hamlet as a threat his throne and at one point even attempts to send him to England and have him killed there. The reason for Claudius to send Hamlet to England is explained when Claudius says "since yet thy cicatrice looks raw and red … thou mayst not coldly set." Cicatrice is a scar or wound and the assumption is that England has recently been wounded by the Danish sword and therefore is more likely to carry out the king's command of killing Hamlet. Also coldly set could mean that they are unlikely to set aside and ignore the fact that Hamlet who is from Denmark is coming to England and with orders to kill him they are not willing to just set this aside and let him live.
Shakespeare also presents Hamlets family to be incestuous especially his mother and uncle who are now married after the death of the old kind. Hamlet seems disgusted that his mother is able to move on so quickly and warm the bed of the old kings with his brother. In Hamlets first soliloquy he says that the time since his father died and his mother remarried is "two months". Ophelia on the other hand said that it has been "twice two months" since the death of the old king and so it could be said that Hamlet shows his disgust over what his mother and uncle have done by marrying by over exaggerating how quickly his father has been forgotten.
In the first soliloquy he also says "so excellent a king, that was to this Hyperion to a satyr". This first part here shows how Hamlet is unfavourable towards his uncle the new king and his father the old king by saying look at what the king once was and look at him now. He compares the two both as family and as the role of king. In both circumstances it seems that Hamlet seems to dislike his uncle and not trust him. "Hyperion to satyr" this again shows how he compares both his uncle the new king and his father the old king by saying that his father is a "hyperion" which is the Greek god of the sun which also uses elemental imagery on how he views his father as the sun which is needed for all life to exist. He then goes on to say that his uncle is a "satyr" which is a grotesque creature, half human and half goat. They were also associated with drunkenness and lechery. This again can be linked back to the fact of Hamlets hate and disgust over what his uncle and mother have done by marrying and sharing the bed that was shared by the mother and the old king at one point.
This dysfunctional family of Hamlets could be argues to be one of the reasons if not them ain reason for Hamlet going downwards in a spiral and eventually resulting in him becoming obsessed with revenge and becoming a fragile state. This can also be backed up by the fact that the whole reason Hamlet even considers revenge in the first place is that his uncle had killed which is also a biblical reference to the story of Cain and Abel. In this story one brother killed the other due to jealousy, greed and to then later on marry the dead brother's wife. This biblical reference seems to go well with what Hamlets uncle has done because he killed his brother due to his own greed for power and the throne of Denmark and he then goes on to marry his brother's wife. Hamlet refers to this as the "greatest sin".
Hamlets obsession to get revenge on his uncle could also been seen as the Oedipus complex which was developed by a man called Freud after watching Hamlet. If his theory was to be used to describe Hamlets actions and his interaction with his mother and his uncle then it would be that Hamlet wishes to get rid of his uncle due to his sexual desire for his mother.
In conclusion Shakespeare's presentation of Hamlets complex family relations is done very well and is intertwined with the overall plot very nicely.