I wonder how Claudius feels about plays in general. When they told him the players were here, did he get a little bit excited, thinking about the monotony of royal life getting broken up with a little drama?
Or did he roll his eyes and think, “Oh dear, I’ll have to go, won’t I? And I’ll have to smile and congratulate them even if it’s terrible. But one must to charitable, mustn’t one. I suppose it might be marginally better than that juggler they brought in a few weeks ago.”
I would guess it’s the former, actually. Hamlet, having watched plays with him before, we presume, since he’s watched them with Polonius, would choose this method of investigation with something Claudius is likely to pay attention to. He would likely choose it because he knows his uncle is particularly likely to get caught up in the wonder and action.
If his uncle were more moved by ballet or ballads, perhaps he’d try those instead.