This is such an interesting question, and a great way of looking at Shakespeare's work. Immediately, I see a connection between Claudius and Macbeth in the way each takes power through the use of violence. Claudius poisons his brother and Macbeth murders Duncan. However, Claudius seems to be the stronger-willed of the two. He seems to have committed the murder on his own, without influence from, or concern for those affected, with merely power in mind. Macbeth, on the other hand, is pressured and influenced by quite a few people to do what he does. His wife plays an extremely influential role in his violent act, as do the witches who give him the prophecy. It is possible that, without these pressures, Macbeth would never have done what he did.
Numerous similarities exist between Shakespeare's Claudius and Macbeth. Both are extremely ambitious. So much so that they both assassinate a reigning monarch to achieve the throne for themselves. Both also, when it becomes necessary, are willing to continue killing to maintain their power. Both show the possibilities of scruples that make them question their own evil actions, although their sincerity is questionable; both may question or feel sorry for what they do/have done because they are worried about the consequences rather than because what they are going to do/have done is wrong.
At the same time, Claudius is more intelligent than Macbeth. Claudius does his own planning, and does it pretty well. Claudius would get away with his assassination if not for the intervention of a ghost, an intervention he couldn't possibly have accounted for. Macbeth needs lady Macbeth to do his planning for him. His evil deeds are discovered once he starts planning his own murders, not when his wife plans a murder for him. Macbeth is also the protagonist of his play and a tragic figure with a tragic flaw, while Claudius is an antagonist and is not a tragic figure.