How did Macbeth kill Duncan in Shakespeare's Macbeth?
Macbeth murders Duncan in his sleep with a dagger in Act II Scene II. He intends to frame the chamberlains for the murder, but is so distraught after the killing that he forgets to leave the dagger behind. Lady Macbeth orchestrates the actual framing of the chamberlains by placing the bloody daggers beside them as they sleep. In Act II Scene III, Macbeth then kills the chamberlains "in a rage" the next morning when he comes upon them in "surprise" with Macduff--this evokes suspicion in the latter, as it is a seemingly vicious overreaction, one which Macbeth tries to support with the claim of being so completely outraged that he could not restrain himself.
There are several reasons for his decision. Macbeth has been told by the witches that he will eventually become king. If Duncan is already king, how can Macbeth become king? Answer: kill Duncan. However, this is Macbeth's first mistake. The witches operate on the basis of fate--it is destined for Macbeth to become king. However, Macbeth takes matters into his own hands (rather than just letting events play out as they will), and that disrupts fate's system. (Or you can argue that it was fated that Macbeth would kill Duncan and that's what the witches were predicting when they said he's become king.)
Lady Macbeth also encourages Macbeth to kill Duncan. She is almost most status and power hungry than he is. She calls his manhood into questions, saying if he was a true man, he would do this. She uses all sorts of feminine manipulation to persuade him into killing Duncan--and Macbeth doesn't want to see like less of a man to his wife.
When the deed is actually to be committed, Lady Macbeth said she would have killed Duncan, but he looked too much like her father when he was asleep. Therefore, it leaves Macbeth to actually do the killing.