How does Macbeth feel about being king but knowing Banquo's children will be kings after him?

It's an interesting and critical question because Macbeth ultimately betrays and murders Banquo, revealing much about the nature of his ambitions and his opinions concerning the prophesy regarding Banquo. For Macbeth, Banquo and his family line represent a threat, which he tries to eliminate through murder.
When speaking about Macbeth, we tend to discuss the subject of ambition (one of his defining character traits, and the source of much of the play's tragedy), and this ambition plays a critical role in his betrayal and murder of Banquo. As we find out in act 3, scene 1, Macbeth is unsatisfied with merely being King. Rather, he wishes to pass the crown down to his children. However, Banquo represents a threat to those ambitions (a factor which Macbeth himself is well aware of). Thus, in order to protect the interests of his own lineage, Macbeth orders the attempted assassination of Banquo and his son in the hopes of eliminating this threat to his own descendants.