I'm going to insert a more controversial opinion here and say that I think the relationship between Hamlet and Claudius is interesting because of their mutual, incestuous interest in Gertrude. There is no denying Claudius' interest, of course. The guy married Gertrude. Hamlet's incestuous desire, though, is often pinpointed through the scene where Hamlet confronts his mother. Of course, this interest can be proven to exist or proven NOT to exist through the text, but it is the idea of that presence that makes "the current lover vs. the interested son" antagonism so very interesting to me. Unfortunately or fortunately, I'm always looking for interesting sexual perversion in literary texts. It's a gift. (Ha!)
I think one of the reasons that the struggle is fascinating is because Claudius and Hamlet are both so intellectually sharp. Both are constantly mentally at war with one another, always trying to keep one step ahead...Claudius to keep Hamlet from discovery, Hamlet trying to achieve discovery.
There is no one else in the court who can match Hamlet's intellect. Polonious believes himself to be sharp but Hamlet runs circles around him. Consider the "fishmonger" scene. Hamlet dazzles Polonious with his words, toying with the man, calling him, essentially a "fool" but Polonious doesn't get it. Instead, he plays right into Hamlet's plan, that is, to convince the old man that he is indeed insane.
Rosencrantz and Guildenstern, Hamlet's childhood friends, certainly have not the wit nor the brains to pull on over on the prince. Once Hamlet gets them alone, it takes just minutes for him to get them to confess their collusion with Claudius.
Horatio is bright but never displays the kind of thinking that Hamlet is capable of.
No, it is Claudius alone who can almost play at Hamlet's level. It is he who first devises the plan to keep him close; then it is he who decides he should be put to death. It is Claudius who devises the evil scheme to have Laertes assure his step-son's death.
The play is a continual back-and-forth between Claudius and Hamlet's mind (and action) games. It is one of the many reasons Hamlet is so endlessly fascinating.
If you look at their relationship from a "mystery" perspective, it is interesting to really consider what each of the characters knows and doesn't know about what the other. For example, Hamlet learns that Claudius is responsible for King Hamlet's murder, but Claudius doesn't have any idea that that is what the problem is with Hamlet. It isn't until the middle of Act 3, after the play within a play that Claudius realizes that Hamlet set him up with that play and that now he has to deal with Hamlet more directly. Even after that scene, Claudius thinks he has total control of the situation because he is just planning to send Hamlet to England on an errand for Denmark. He is completely protecting himself in the eyes of Denmark. He has no idea that Hamlet has read the letter dictating his death and that he returns to Denmark with that knowledge.
On the other hand, it is fun to watch Hamlet taunting everyone around him with his "crazy act" and his broad hints about what he knows and suspects of everyone's behavior. We know that Hamlet is playing games even through Act 4 and we enjoy watching the jokes go right over the other characters heads, like Polonius, or that the jokes are very rude and perhaps a bit frustrating as with Hamlet's interactions with Claudius.
We also watch what Hamlet reveals to Horatio and to his mother about what he is planning, and look forward to seeing how the action of the story will play out. We know that Claudius will get what he deserves, but we wonder what the cost will be.