How are Duncan and Macbeth related?
There is no blood relation between Duncan and Macbeth: Duncan is the King of Scotland and Macbeth is one of his Thanes. That said, there is undoubtedly a close relationship: Duncan does call Macbeth his "worthiest cousin" (though "cousin" in Shakespeare just means someone close to you - can be either a blood relation, or a friend).
And what Duncan says about him at the end of Act 1, Scene 4 refers to him as "kinsman" - somebody loyal, somebody "of kin":
Let's after him,
Whose care is gone before to bid us welcome:
It is a peerless kinsman
So no blood relationship - but an extremely close, loyal friendship: and of course, the loyalty due from a subject to a king.
They are actually cousins. That is one of the reasons that Macbeth does not want to kill Duncan, that and vengeance, Loyalty, Hospitality, Religion and that it will unbalance nature. A good version to read it the Cambridge version it helped me alot this year in 8th grade.
They are cousin. <<<True answer>>>
There is a historical basis for the relationship between the two. Shakespeare uses Raphael Holished's Chronicles of England, Scotland, and Ireland for some of the information.
"When King Malcolm II of Scotland died in 1034, his last command was that the throne should pass to his oldest grandson, Duncan. This last request went against the Celtic tradition of succession, which stipulated that the inheritance of the throne alternate between different branches of the family, and simultaneously cut out another grandson, Macbeth, from the line of inheritance. Macbeth nevertheless pursued his claim to the throne as prescribed by tradition." see full quote below
So even though in the play they are not actually related, Shakespeare was setting up a conflict that had existed based on a convoluted blood line to the throne.