Why was Macbeth right to kill King Duncan?
I agree that the proper question is "Was Macbeth right to Kill Duncun?" and that the only proper answer is NO. This is why the play pretty much begins with the murder: Shakespeare is more interested in what are the consequences of an evil act upon its perpetrators than in a story about killing a king.
However, you asked what you asked, so I wish to address that as well. There are those who argue that Duncan was not an effective king. Act I, scene 2 makes it quite obvious that Duncan would have lost the war if not for Macbeth's courage in battle. Now, one of the puzzles of reading some of Shakespeare's plays is that the king's son is not always the one to inherit the kingdom. Hamlet is 30 years old when his father dies, yet for some reason he has not been crowned king; instead his uncle has. And in Macbeth, Malcolm is in line to become king only because Duncan has announced that is his decision. It could be that in these early kingdoms, the king was required to be the toughest guy around. This idea is expressed by Isaac Asimov in Asimov's Guide to Shakespeare:
"Ideally, the strongest and most competent member of the royal family should be on the throne. Duncan was soft and gentle, but he was also negligent and inefficient. He was not a king for those hard times" (volume 2, page 156, or page 122 of the attached PDF).
Surely, Macbeth was angered to hear that Duncan had chosen Malcolm. This would imply that this choice was not to be automatically presumed. Similarly, Duncan has placed trust in people who proved untrustworthy, particularly the Thane of Cawdor. And in naming Macbeth the new Thane of Cawdor, Duncan repeats the error! What's more, Duncan is told what terms of surrender were agreed to by the vanquished army's leaders. He does not appear to have dictated the terms or played much of a role in the negotiations. So there are those who believe that Macbeth deserved to be king, and as a strong warrior who has almost single-handedly won the battle, that he might feel entitled to be the king.
I cannot see that it was right in any way for Macbeth to kill Duncan. The King was his friend, his patron, his feudal lord, his trusting guest, and a well-loved monarch. Macbeth himself acknowledges in his soliloquy in Act 1, Scene 7 that he has no legitimate right to go through with the assassination:
He's here in double trust:
First, as I am his kinsman and his subject,
Strong both against the deed; then, as his host,
Who should against his murderer shut the door,
Not bear the knife myself. Besides, this Duncan
Hath borne his faculties so meek, hath been
So clear in his great office, that his virtues
Will plead like angels trumpet-tongued against
The deep damnation of his taking-off,
And pity, like a naked new-born babe,
Striding the blast, or heaven's cherubin horsed
Upon the sightless couriers of the air,
Shall blow the horrid deed in every eye,
That tears shall drown the wind. I have no spur
To prick the sides of my intent, but only
What Macbeth says about being Duncan's host is especially interesting. Both Macbeth and his wife see Duncan's overnight stay in their castle as a golden opportunity to kill the old man. Yet at the same time they are morally obligated as host and hostess to take good care of him. Actually they both must know is it wrong but want to do it anyway because they are both so insanely ambitious. They cannot foresee the consequences of killing the king. Macbeth automatically loses his good reputation and is regarded widely as a villain and tyrant. People hate and fear him. They don't want to cooperate with him. They don't want to fight for him. Macbeth has actually killed the man who gave him such a good reputation in the first place. Macbeth does not say so in exact words, but he knows he has sold his soul to the devil ("Given [it] to the common enemy of man") and that he has committed an unpardonable sin.
To answer your question as it was written, in the minds of Lady Macbeth and even Macbeth himself, killing King Duncan is, indeed, the "right" thing to do in their path to power.
In Act I, Scene 5, Lady Macbeth speaks aloud after reading her husband's letter informing her that he has been made Thane of Cawdor and that the witches have predicted he will also become king:
Hie thee hither,That I may pour my spirits in thine earAnd chastise with the valor of my tongueAll that impedes thee from the golden round,Which fate and metaphysical aid doth seemTo have thee crowned withal. (1.5.15-20)
The Prince of Cumberland! That is a step
On which I must fall down, or else o'erleap
Let not light see my black and deep desires....(1.4.50-53)
I do not believe that Macbeth was right in killing Duncan. It is murder no matter how you look at it. Some readers blame Lady Macbeth for the murder of Duncan. She did goad her husband into doing the deed by calling him all kinds of names and questioning his manhood. All of that may be true, but it was still Macbeth who plunged the knife into the sleeping Duncan. Macbeth didn't even have the courage to challenge Duncan into some kind of a fair fight. I believe the previous post is correct in asking the question "was Macbeth right to kill the king?" If that really is the question, I stand by my thoughts that "no", Macbeth was not right.
If your question really is "Why was Macbeth right to kill king Duncan," then I'll try and provide a reason. Macbeth was right to kill Duncan because it was the simplest and fastest way to attain the throne. As a Thane, Macbeth knows that there is a remote chance for him to be king. He gets the prophecy from the witches, and the seeds of temptation have been planted. Upon receiving the title of Thane of Cawdor, Macbeth knows that he is one step closer to being king. At this point, Macbeth can wait it out and see if nature takes its course, or he can kill Duncan and immediately be king.
Occam's razor says "the simplest solution is usually the correct solution." According to Occam's razor, Macbeth is right in killing Duncan; it offers the simplest solution to Macbeth's problem. Macbeth's question = "How can I be king?" Simplest solution = kill Duncan. Macbeth was right, because it solved his dilemma with the fewest possible steps and variables.
I believe that the real question should be "was Macbeth right to kill the king?" The witches' prophesy says that Macbeth will become the King. Their power operates on fate. Macbeth takes fate into his own hands and decides to kill the King so that he can fulfill the prophesy himself. The question at hand now is whether or not the Witches knew that Macbeth would take matters into his own hands and that is how they knew he would become King, or if Macbeth got ahead of himself and did not allow fate to take its course with his kingship.
Killing the King made Macbeth the rightful heir which fulfilled the prophesy of becoming the King himself. Furthermore, it is really Lady Macbeth who pushes Macbeth to kill the King. One could say she is more greedy than he is. She plots the death and pushes Macbeth to complete the task by questioning his manhood and her womanhood.