What does Hamlet stalling killing Claudius say about Hamlet's approach to revenge? Does his approach change the genre of the play from merely a...

I don't know that Hamlet has ever been merely a "revenge" play. While avenging his father's death is Hamlet's motivating impulse for the plot, the play has always been about the internal struggle Hamlet wrestles with, and the larger thematic implications that result.

Hamlet vows on several occasions to kill Claudius, and his determination seems sincere. He tells his father's ghost, "thy commandment [to kill Claudius] all alone shall live within the book and volume of my brain," and he promises that his "thoughts be bloody or be nothing worth." Nonetheless, he first decides to test the ghost's word by setting up the play-within-the-play, and later passes up the opportunity to kill Claudius while he prays, choosing not to "take him in the purging of his soul." This brings the ghost back to remind Hamlet to fulfill his "almost blunted purpose." Hamlet still does not act, instead choosing to procrastinate and make excuses.

What this says about Hamlet isn't so much about his "approach to revenge" as it is about his reluctance to murder his mother's lover. Oedipal interpretations aside, doing so would mean Hamlet would, in all likelihood, estrange himself from his mother, to whom he is deeply devoted. Worse, it would mean Hamlet himself would assume the throne and all its attendant responsibilities. Hamlet's a college student who wants to return to Wittenburg and resume his studies. The idea of ruling Denmark and filling the kingly shoes of his great father is probably pretty intimidating. Moreover, killing Claudius is an act Hamlet presumably has difficulty stomaching. He's only able to kill Polonius when he's hiding behind an "arras," where Hamlet doesn't have to face him. Murdering Claudius would most likely be a more intimate act, and one that the creative and romantic Hamlet may lack the willpower to commit. His contemplation of suicide in both of his early soliloquies suggests Hamlet may prefer to orchestrate his own death than his uncle's, so conflicted is he about his responsibility to his father and country.  

Ultimately, Hamlet doesn't kill himself, either. Instead, he feigns madness, orchestrates a play, toys with Ophelia's heart, aggravates Polonius, and duels Laertes. Hamlet, quite simply, doesn't know what to do. In spite of the obligation he has taken on, Hamlet seemingly lacks the resolve to fulfill it. Thus, the play is more about the existential questions he raises as he wrestles with his conscience: do our actions on earth even matter? Is death preferable to life? Are we merely waiting to become "clay [to] stop a hole and keep the wind away"? The revenge aspect of Hamlet is merely the vehicle by which Shakespeare introduces his larger and more far-reaching themes, and this is at least partly responsible for its timeless and enduring appeal.