In Shakespeare's Macbeth, two main events lead to others in the play suspecting Macbeth of treachery.
First, Macbeth kills the two grooms that could have been witnesses of Duncan's murder. In Act 2.3 Lennox states that the grooms seem to have done the murder, since their daggers and hands and faces were covered in blood. The thanes present would of course want to interrogate them, but Macbeth makes that impossible, announcing:
Oh, yet I do repent me of my fury,
That I did kill them. (Act 2.3.103-104)
Though no one else says anything at the time, Macduff realizes the problem with Macbeth's killing of the grooms:
Wherefore did you so? (Act 2.3.104)
Macbeth tries to explain his actions and play it off, so to speak, but Macduff is not convinced. When the time comes for Macbeth's coronation as king, instead of going to witness and celebrate with Macbeth, Macduff instead goes home "to Fife (Act 2.4.37), further revealing his suspicions. And, of course, Macduff turns out to be directly involved in Macbeth's downfall.
Second, the killing of Banquo and the attempted killing of Fleance then lead to widespread suspicions. In Act 3.6 Lennox and an unidentified Lord speak sarcastically and ironically about the coincidences of Duncan's and Banquo's deaths, as well as the deaths of the two grooms. By this point, too many murders have occurred for others to not suspect Macbeth of treachery.
Adding to the murders, is the fact that, apparently, Macbeth has become a tyrant. Lennox refer's to the "tyrant's feast," and the Lord refers to Macbeth as "this tyrant." The two hope Macbeth will be overthrown, so that
...we may again
Give to our tables meat, sleep to our nights,
Free from our feasts and banquets bloody knives,... (Act 3.6.33-35)
Thus, Macbeth's tyranny has made others in the play discontented, and the string of murders has made them suspicious.
Ironically, even Lady Macbeth almost immediately adds to the suspicion after Macbeth kills the grooms. Macbeth veers from her plan when he kills the grooms and Banquo, as stated. He'd have been much better off following her directions. But even she reveals evidence of their guilt, even though Banquo seems to be the only one who catches it. When she is told of Duncan's death, she responds:
What, in our house? (Act 2.3.84)
And Banquo understands the inappropriateness of her response and rebukes her:
Too cruel anywhere. (Act 2.3.84)
Banquo understands that there is something warped about a woman whose first thought after hearing of a king's assassination is that it reflects badly on her hospitality. Banquo never acts on his suspicions, but they are certainly present in his mind.