Even though she does not kill Duncan with her own hands, Lady Macbeth is the driving force behind her husband Macbeth, who does the deed.
In Act I, Scene 3, Macbeth first gets the idea that he will one day be King when the witches address him,
"All hail, Macbeth, that shall be King hereafter!" (line 53).
In Act I, Scene 5, Macbeth informs his wife of what the witches have said in a letter. He writes that they have proclaimed,
"King that shalt be" (line 9),
and tells her,
"This have I thought good to deliver thee, my dearest partner of greatness, that thou mightst not lose the dues of rejoicing, by being ignorant of what greatness is promised thee" (lines 9-12).
Lady Macbeth is impressed, and, in her greed, immediately begins to doubt that Macbeth has the guts to do what it takes to make the prophesy come true, thinking,
"I do fear thy nature; it is too full o' the milk of human kindness to catch the nearest way" (lines 14-16).
When Macbeth returns home in Act I, Scene 7, Lady Macbeth finds that her husband has indeed thought about killing Duncan in order to force the prophesy to come true quickly, but he is wavering in his resolve. Lady Macbeth goads him on, telling him essentially that he was more of a man when he dared to think of taking action and instigating Duncan's murder;
"When you durst do it, then you were a man" (line 55).
Now that Macbeth is mired in uncertainty, she says with extreme harshness that if she were in his place, she would have
"...pluck'd (her) nipple from (her nursing babe's) boneless gums, and dash'd the brains out" (lines 63-64)
rather than back out of what he has planned. Lady Macbeth then takes things into her own ruthless hands, telling Macbeth to
"screw your courage to the sticking-place" (line 68).
She then lays concrete plans to have Macbeth murder Duncan while he sleeps that night, so that Macbeth might take his place as King.