Both Macbeth and Claudius of "Hamlet" are kings who got to be kings because they not only killed the current king, the king they killed was a relative. Macbeth kills his cousin, King Duncan. Claudius kills his brother, King Hamlet. Macbeth killed out of "vaulting ambition" as he identified it in Act 1, sc. 4. Claudius killed because he wanted the queen, Gertrude, the throne, and because of his "ambition" which he tells us as he is in the chapel in Act 3, sc. 3. Both kings are willing to commit further murders to cover their initial crimes. Macbeth has Banquo killed because Banquo suspects him (Act 3, sc. 4). Claudius tries to have Hamlet killed (Act 5, sc. 2) and doesn't do much to stop Gertrude from drinking the poisoned wine in Act 5, sc. 2. Both men love their wives, though those wives are quite different. Lady Macbeth knew her husband committed murder, even helped him and planned it. Gertrude seemingly did not know Claudius killed her first husband.
I agree with the above comments.
I think an important contrast is that Claudius murdered willingly, independently and nobody else was involved, whereas Macbeth was very unwilling to murder Duncan and was reluctantly 'following orders' from his bold wife.
Both Kings are tortured by the guilt of their murders. Both feel the weight of their 'sin' and fail to find absolution. Both are dragged deeper into crime to hide their guilt and protect their stolen crown. Both pay for their crimes with their lives and the lives of their loved ones.
Macbeth is more of a warrior, whereas Claudius is more like a politician and courtier. (There was a real King Macbeth of Scotland in about the 11th century but his story is not the same as Shakespeare's play)