In Macbeth, how does killing Banquo connect to the killing of Macduff's family?
Concerning Shakepeare's Macbeth, the ordering of killings demonstrates the extent to which Macbeth will go in order to maintain and enhance his power--sort of.
Ordering Banquo's murder makes strategic sense for Macbeth, because Banquo knows about the witches' predictions. Banquo suspects Macbeth of murdering Duncan and Macbeth suspects he suspects, so to speak. Killing Banquo is a strategic move. Ordering the death of Fleance, however, demonstrates how Macbeth's ambition has grown. Before killing Duncan, all Macbeth dreamed of was being king. He had no thoughts of creating a dynasty by having his heirs continue as kings of Scotland. Once he has the crown, though, it's no longer enough. He orders Fleance's death in an attempt to open the way for him to create a dynasty. He tries to enhance his power.
Ordering the slaughter of Maduff's family, however, does not improve Macbeth's chances of holding on to power, nor does it enhance his power. It serves no strategic purpose. Killing Macduff would serve a strategic purpose, but slaughtering his family does not. This slaughter is more personal. Macduff refuses to give loyalty to Macbeth, snubbing Macbeth by not attending his coronation or his feast. When Macbeth finds out that Macduff has traveled to England to join Malcolm and seek help from the English, he lashes out at Macduff by ordering the slaughter of his family. Macbeth can't get at Macduff, so he lashes out at his family instead. This demonstrates, not Macbeth's ambition or its growth, but his pettiness and ruthlessness and callousness. It's almost like Macbeth is throwing a sophisticated temper tantrum--very dangerous when we're talking about a man with great power in his kingdom.
In short, then, the killings you ask about are connected in that a tyrant orders them. They also, by the way, reveal Macbeth doing his own planning (not involving his wife, who is the better planner) since murdering Duncan. But the murders are different in the purposes they serve and by what motivates them.
After Macbeth slays Banquo, his ghost appears at Macbeth's coronation banquet. After being unhappily entertained by Banquo's unexpected appearance, Macbeth notices Macduff's absence. He goes again to the weird witch sisters who warn him to beware of Macduff, the Thane of Fife. The witches vanish after imparting this and other disheartening news--it looks like Banquo was correct in suggesting that the witches were giving juicy tid-bits to entice for the purpose of leading into real harm--and Lennox approaches Macbeth to tell him that Macduff has fled to England; has turned against him; and has joined forces with Malcolm. This enrages Macbeth (and frightens him) who in retaliation orders that Macduff's wife and children, his whole family, be slain, leaving no survivor. Banquo connects Macbeth to Macduff because Banquo's ghost at the banquet disturbs Macbeth's mind even further than events have already done, which unhinges it to the point of rage and senseless retaliation.
Killing of Banquo is related to killing of Macduff's family as it shows Macbeth's change in his mindset about murdering other people. It shows how a tentative Macbeth, who was thinking whether or not to kill Duncan, starts to take hasty decisions, wiping out Macduff's family.