Fundamentally, the reason for act four being the falling action is with the killing of Polonius leading to Hamlet being sent away, Ophelia's death as well as Laertes returning from France to take his revenge, this all plays a significant role in how the play play's out.

Gertrude O me, what hast thou done?

Hamlet Nay, I know not. Is it the king?" (Act 3, scene 4, line 25-27). As Hamlet kills Polonius, he is committing a violent crime which leads to ongoing conflicts with the King and the son of Polonius, Laertes. Hamlet's actions are what causes Claudius to realize how Hamlet is a threat to him, thus representing the quarrel between the two, which eventually leads to the falling action of the play. In addition, the third act presents the question to the audience if Hamlet may have actually gone insane. After Hamlet murders Polonius, he is quick to deflect the situation back to his mother and her betrayal to his father, "A bloody deed? Almost as bad, good mother, As kill a king and marry with his brother." (Act 3, scene 4, line 29-30) which leads to Hamlet seeing his father's ghost once again. Except this time, Hamlet is the only one able to see and speak to the ghost while Gertrude cannot. Gertrude determines that her son is mad for seeing a ghost, which also poses the question to the audience whether this ghost began to appear only in Hamlet's mere imagination. 

In the third act, the conflict between Hamlet and Claudius becomes the principal source of appearance versus reality as Hamlet  is continuously trying to appear as though he has no problems with Claudius while he is actually plotting to kill him. For instance, in act 3, when Claudius asks how hamlet is doing, he responds by saying: "Excellent, i’faith, of the chameleons dish. I eat the air, promise-crammed. You cannot feed capons so.” (Act 3, scene 2, line 86-87). It is clear that to Claudius,  Hamlet seems fine, however dramatic irony is portrayed as the audience knows this is not the case. 

Throughout act 4 is when Claudius becomes the main portrayal of deception as he is continuously hiding his true intentions. As Claudius claims he is sending Hamlet to England to be healed from his supposed madness, he is actually sending him to be killed in England. While Claudius has already been planning to have him killed, it appears that he is sending Hamlet to England for Hamlet's benefit: “Hamlet, this deed, for thine especial safety- Which we do tender as we dearly grieve For that which thou hast done- must send thee hence With fiery quickness” (act 4, scene 3, line 40-42). 

Appearance versus is present in act 5 as Laertes is also planning to kill Hamlet during the sword fight however he is making it seem as if he does not want to seek revenge for the murder of his father: "I am satisfied in nature,

Whose motive in this case should stir me most

To my revenge. But in my terms of honor

I stand aloof, and will no reconcilement"( act 5, scene 2, 232-234)

King Claudius is evidently the antagonist of the play due to his selfishness and his manipulation of situations to match his own benefits. The audience immediately perceives Claudius to be the “villain” in the play as it is revealed he is the one to kill the past king which happened to also be his own brother. However later on in the play, Claudius seems to be somewhat innocent as while everybody is indicating that Hamlet has gone insane, Claudius is more so understanding of Hamlet’s feelings: “Love? His affections do not that way tend. Nor what he spake, though it lacked form a little, Was not like madness” (act 3, scene 1, line 162-164). This plays out to be ironic as in that moment, Hamlet’s state of “madness” did not affect Claudius, therefore Claudius does not seem eager to send Hamlet off to England. However once Hamlet’s madness puts Claudius’s power at risk, Claudius is quick to convince everyone that Hamlet has gone completely mad and must be sent to England immediately: “His liberty is full of threats to all- to you yourself, to us, to everyone” (Act 4, Scene 1, line 14-15). This exemplifies the way Claudius begins to manipulate a situation into making it seem as if it is of everyone's best interest to do something, when the only one gaining the real benefits is him. In the third act, while Claudius is confessing to his murder of King Hamlet, it appears that he genuinely feels guilty: O, my offence is rank, it smells to heaven; It hath the primal eldest curse upon't, A brother's murther! Pray can I not, Though inclination be as sharp as will. My stronger guilt defeats my strong intent” (Act 3, scene 3, line 37-41). However at the same time Claudius’s selfishness is revealed as although he feels extreme guilt, he is not willing to give up the benefits of the past king which included power as well as his wife: “May one be pardon'd and retain th' offence?” (Act 3, scene 3, line 57). 

The actions Claudius starts to take in act four by telling Laertes that Hamlet was the one to kill his father, Polonius are what ultimately sets himself up for his own death at the end of the story. Claudius’s motive is to gain Laertes on his side: “Now you’ve got to acknowledge my innocence and believe i’m your friend, since you’ve heard and understood that the man who killed  your father was trying to kill me” (act 4, scene 7, line 1-5)  in order to help him kill Hamlet,  leading to the two of them creating a plan to kill Hamlet - which backfires and deflects on themselves, killing both of them. As Claudius is passionate about killing Hamlet for his own selfish needs, his main focus was what he would gain out of Hamlet’s death instead of thinking thoroughly on how he may handle the situation in another manner that would have not lead to the death of anyone. Finally exemplifying how by constantly only thinking of himself, he was the cause of his own death.