Macbeth's cover story for the death of Banquo is that he was killed by his son Fleance. We do not know if Macbeth spread this story directly, but in Act III Scene 6 Lennox makes a sarcastic reference to it:
And the right-valiant Banquo walk'd too late;
Whom, you may say, if't please you, Fleance kill'd,
For Fleance fled: men must not walk too late.
Thus, the partial failure of Macbeth's plan -- in that Fleance, who was supposed to share his father's fate, instead escaped -- also made Fleance a natural suspect in the murder.
The accident of Fleance's escape and flight makes the death of Banquo resemble that of King Duncan in that in both cases, Macbeth arranged the murders (personally or through intermediaries), and in both cases the surviving sons of the murdered men fled and thus brought suspicion on themselves, which Macbeth could exploit to help hide his role. It was a weak explanation at best, its weakness driven home by Lennox's narration, which explicitly links the two cases:
Who cannot want the thought how monstrous
It was for Malcolm and for Donalbain
To kill their gracious father? damned fact!...
....and I do think
That had he Duncan's sons under his key--
As, an't please heaven, he shall not--they
What 'twere to kill a father; so should Fleance.
Lennox had initially thought that the king's servants had murdered him (Act II, Scene 3), and Macbeth had later put it about that this had been done on the instigation of the king's sons, who had fled. However, it seems to have been a case of "once bitten, twice shy" with Lennox. He cannot believe that those who happen to stand in Macbeth's way have been stricken with such a convenient epidemic of parricide, and is forced to look for other motives and other murderers than he might first have thought.