Hamlet's delay (and the actions of others; namely, Claudius') leads to his tragic fall. That fall (fault) is what leads to the deaths of Ophelia, Gertrude, Polonius, Laertes, Claudius, and himself. Polonius' death was an accident as was Gertrude's (she drank from the cup poisoned by Claudius). If Hamlet had killed Claudius immediately, the rest of the bloodshed would probably not have occurred.
This is one of the most popular debates about Hamlet. Why does he delay? Hamlet is too philosophical, and he knows this. He longs to be more like Fortinbras who acts as much as he thinks. Also, since Hamlet is so grief-stricken at his father's death and upset with his mother for marrying Claudius so quickly, when he does decide to avenge his father, he wants to do so in the most dramatic way possible: to make a statement. This causes further delay. For example, he has an opportunity to kill Claudius when he is alone; but since Claudius is praying, Hamlet reasons that this will not be complete revenge because he is praying; killing Claudius would send him to heaven.
And am I then revenged
To take him in the purging of his soul,
When he is fit and seasoned for his passage?
In delaying and stretching out the drama, Hamlet is simply trying to enact his revenge in the best way possible. But over the course of that delay, he accidentally kills Polonius. He also alienates Ophelia and her depression leads to her suicide. Laertes becomes his enemy because of this. Then Laertes and Claudius find themselves allies against Hamlet. They plot to get rid of Hamlet either with a poisoned sword in a duel or the backup plan of having Hamlet drink from a poisoned cup. Gertrude drinks from the cup.
As a moral philosopher debating the meaning of life, revenge, and/or suicide, and in the wake of his father's death (and seeing his ghost), Hamlet's philosophical pontifications are understandable. However, his delay does inadvertently lead to these other tragedies. That is not to say they are all his fault. Claudius is the instigator and continues to conspire against Hamlet, thus Claudius also extends the potential for tragedy. So, Hamlet is only partly responsible for his tragic fall. Claudius set the wheels of tragedy in motion. Hamlet, attempting to stop those wheels in the most significant way, kept those wheels going because of his delay. As the wheels kept going, tragedy kept occurring.