Why is Hamlet considered to have problematic subjectivity?

Yes, there are problems with subjectivity in Hamlet the play, in Hamlet the character, but mainly subjectivity problems with critics writing about the character of Hamlet.

The role of Hamlet is just that.  It's a role.  As Eliot points out in his wonderful essay, "Hamlet and His Problems," critics tend to focus entirely on Hamlet as a psychological puzzle to solve, when psychology was not even a discourse in Shakespeare's era. Eliot calls Hamlet, the play, an "artistic failure," and says not even Shakespeare understood the character: that he superimposed his own language over an already constructed plot, Kyd's The Spanish Tragedy and Ur-Hamlet.

Critics make the mistake, according to Eliot, of interpreting Hamlet in their own image; hence, Coleridge's Hamlet is essentially Coleridge.  Where's the subjectivity there?  Can one speak of subjectivity when discussing an objective thing: a role, a character?  Transference is inevitable.

Hamlet is, more or less, an actor playing an actor playing an actor.  He's not sure who he is, but he knows who he wants to be: an actor.  He comes alive, not when he's been assigned revenge by his father's ghost, but when the traveling actors arrive.  He relishes his role as The Mousetrap's producer/director.  He loves theater, not so much life outside the theater.  So, is there subjectivity in metadrama / metatheater?

You should check out Hamlethaven.com.  I cite the following article:

Amtower, Laurel. “The Ethics of Subjectivity in Hamlet.” Studies in the Humanities 21.2 (Dec. 1994): 120-33.

Amtower says:

the murder of Old Hamlet “is never known in its actuality, but is instead delivered as information, filtered through the suspicious perspectives of the characters, and acted upon accordingly” (124). After gaining “information” about his father’s murder, Hamlet responds to the call for revenge by attempting to “justify the task within the theological and political framework that structures not only his ethical sensibilities, but his very sensibilities regarding who and what he is” (125). “Hamlet is thus placed into a subjective crux within which intersect the exclusive values which frame his very being” (125). But by “believing he acts for a higher agency” (e.g., the Ghost/father) and thus “dismissing the claims of his own integrity,” Hamlet “begins to reinscribe the entities and relationships around him into narratives and texts, to be negotiated and interpreted according to his own absolute gloss” (126). For him, absolutes “become fluid,” and “life is nothing but a language game” (126). Unfortunately, Hamlet is “not just a player of games comprised of words and deceptions, but a product of these games” (128). He feigns madness and manipulates The Mousetrap, all language-based methods, to extract truth from others—but egotistically neglects the fact that “the ‘truth’ he seeks might well be a product of his own discursive devising” (129). Leaving behind humanity and morality, he “appoints himself ‘scourge and minister’” (131) and “perverts the discourse of religious dogma in the pursuit of selfish ends, for the subject at the end of this play is a tyrant, using the discourse of power to justify his abandonment of individual ethics” (132).

Hope this helps...