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 all costs, even to the point of murdering Hamlet'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.