If we understand the term "nemesis" to mean death (a nemesis is an enemy and death, to most people, is a chief enemy), then it is primarily Claudius's ambition and treachery that acts as the nemesis in the play, leading to multiple deaths.
Claudius wanted to be king at...
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all costs, even to the point of murderingHamlet's father, the king. This sets into motion a series of actions that culminates in the deaths of Claudius, Gertrude, Laertes, and Hamlet.
The sequence of events is as follows: Claudius kills King Hamlet. The ghost of King Hamlet visits Prince Hamlet and tells him to avenge this death. Prince Hamlet takes time determining that Claudius really is guilty. Then he kills Polonius by mistake, thinking he is Claudius. This brings Laertes back to Denmark, ready to kill Hamlet to avenge his own father's death.
Claudius gets the hot-headed Laertes to fight Hamlet with a poison-tipped rapier. Both Laertes and Hamlet are stabbed with it, so they are doomed to die. Gertrude dies when she drinks poisoned wine meant by Claudius to be drunk by Hamlet. Before he dies, Prince Hamlet kills Claudius.
All these deaths can be traced directly back to Claudius. However, each character also had a weakness that was their own personal nemesis or enemy: Hamlet was too indecisive, Gertrude was too weak and willing to go along with Claudius, and Laertes was too hot-headed. These factors also contributed to their deaths.
I am not entirely sure what you mean by "nemesis" but one of the definitions is "something that a person cannot conquer or acheive". If this is the case, then Claudius never achieves a true kingship (he cheats and murders to gain the throne, only to be killed himself), Gertrude loses her status as a respected queen; Laertes loses his sister and father and is doomed to a life of regret for leaving them both to pursue his own agendas; Hamlet avenges his father's death, but loses his mother and the fate of Denmark to Fortinbras in the process. "Nemesis," therefore, in this definition, meets the criteria of tragedy...and Hamlet is one of literature's finest all-round examples of the term.
Explain how nemeses overtook Claudius, Laertes, Gertrude, and Hamlet.
Prince Hamlet is Claudius's true nemesis. Toward the beginning of the play, Prince Hamlet was visited by his father's ghost and was instructed to avenge his death. Prince Hamlet is not only furious that Claudius assassinated his father but is also disgusted that he has married Gertrude immediately after usurping power. For the majority of the play, Hamlet hesitates to kill Claudius until the final scene. Hamlet eventually stabs Claudius with Laertes's poison-tipped sword and forces him to drink the poisoned wine.
Laertes is initially Prince Hamlet's nemesis. Laertes is manipulated by Claudius and seeks to avenge the death of his father and sister by killing Prince Hamlet. Laertes learns that Prince Hamlet has killed Polonius and believes that he is also responsible for Ophelia's suicide. However, Laertes does apologize and forgive Hamlet before he dies. Similar to Gertrude, Laertes is used as one of Claudius's pawns and tragically dies at the end of the play, when Hamlet stabs him with his own poison-tipped sword.
Hamlet could be considered Gertrude's nemesis, in a sense, but he has been given strict instructions by his father's ghost to not harm her. Hamlet is simply disgusted that Gertrude quickly married Claudius after his father's death. She is also unknowingly a pawn in Claudius's plot to ensure his authority and kill Hamlet. Gertrude dies after accidentally drinking the poisoned wine meant for Hamlet during the fencing match in act 5, scene 2.
Hamlet's primary nemesis is King Claudius, who is wary of him and wants him dead in order to cement his legacy and ensure stability throughout his kingdom. Claudius sends Hamlet to England, where he gives the English authorities instructions to kill his nephew. King Claudius uses Laertes to kill Hamlet by allowing Laertes to fence with a poison-tipped sword. Claudius also has Hamlet's wine poisoned to ensure that he will die, which backfires after Gertrude accidentally drinks from the cup. Tragically, Hamlet dies after being stabbed by Laertes during the fencing match.
Explain how nemeses overtook Claudius, Laertes, Gertrude, and Hamlet.
A nemesis is generally a foe who seeks revenge; in a broader sense, it's any enemy. I'm not sure these characters all have a true nemesis, but they each die because of someone else's actions.
Laertes - Hamlet is the one who kills Laertes, but he absolves Hamlet of guilt. He understands, as he's dying, that Claudius fed his anger against Hamlet and used him to try to further his plans to kill Hamlet.
Claudius - He's only one of two in this play who does die at the hands of a true nemesis--Hamlet avenging his father's murder.
Gertrude - She dies for much the same reason as Laertes, a casualty of Claudius's plot to kill Hamlet.
Hamlet - While it is Laertes who inflicts Hamlet's fatal wound, Hamlet dies due to the scheming of Claudius. Claudius is his true nemesis, in that the King has plotted and schemed to have him killed.