Well, according to the stage directions in the play, Hamlet enters after Claudius gives his soliloquy, which is intensely spiritual and focused on his soul and the afterlife. Observe:
O, my offence is rank it smells to heaven;
O, what form of prayer
Can serve my turn?
O wretched state! O bosom black as death!
O limed soul, that, struggling to be free,
Art more engaged! Help, angels! Make assay!
You can't have Hamlet step on those lines with an entry. They are full of apostrophes to pain and suffering, and to have another body on stage will diminish their power and relevancy.
Hamlet enters and says his monologue, so the audience should be watching and listening to it, for it is a response to the soliloquy. You can't have the audience's eye drawn to Claudius acting like he knows Hamlet is there. It's just too hoaky.
Plus, there's not enough time. Claudius only has a couple of lines after that, and then he's gone. Not to mention that when Claudius says, "My words fly up...," he's talking about the silent ones in prayer, so he's obviously busy praying. His praying is dramatic irony enough: the main thing is that we know Hamlet is there, not him. You can't have him praying and noticing Hamlet. Too busy.
The scene is about spiritual crisis for both Claudius and Hamlet, not about the closeness of two actors. It's about Claudius worrying about his soul, maybe for the first time. Hamlet, too, is worried about Claudius' soul: he doesn't want to send him to heaven, only hell. He realizes that it's spiritually not the right time to kill, which is ironic and funny.
Let the audience focus on the words, not some non-verbal action on stage. Shakespeare's all about the language, not the action, and certainly not subtext and subtleties. I wouldn't diminish those words for anything. That's what a movie adaptation is for. You might be able to do it with a close-up, but not on stage, not even in the round. Too much can go wrong.