Symbolism, Imagery, Allegory
Hamlet first comes up in Act 1, when Arkadina is impatiently waiting for Konstantin's play to begin. Totally incapable of sharing the spotlight, the actress quotes a bit of Gertrude (Hamlet's mother) to get some attention. Konstantin does her one better. He mocks her romance with Trigorin by referencing the moment when Hamlet criticizes Gertrude's affair with Claudius:
Nay, but to live
in the rank sweat of an enseamed bed,
stewed in corruption, honeying and making love
over the nasty sty. (1.89)
You can learn more about the Hamlet/Gertrude relationship here, but what's important here is its Oedipal nature—the sexual undercurrent between mother and son. Think about it: Konstantin plays the "she loves me, she loves me not" game about Arkadina, not Nina. He has lots of knock-down-drag-out arguments with Arkadina, followed by kissy-kissy making-up scenes:
Konstantin: The last few days, I've loved you the way I did when I was little. Totally, tenderly. You're all I've got. (3.65)
And more than anything, he can't stand the way she treats Trigorin. The dysfunctional mother/son relationship—along with the rejection from Nina—helps drive Konstantin to suicide.