For most critics, Hamlet is, first and foremost, a revenge tragedy. How does this view ignore the philosophical aspects of the play?

To limit an assessment of Hamlet as a revenge tragedy is indeed short-sighted. Throughout the play, Hamlet struggles to understand himself and the meaning of life, thus a more thoughtful theme of existentialism emerges.  In his examination of life and death, Hamlet observes that man ultimately becomes food for worms, and he delivers this insight in a witty and punishing remark to Claudius to remind him of death's equalizing power.  Yet in the same act (Four), Hamlet considers the idea that God wouldn't give man profound intellect and powers if He didn't intend for man to make profound use of them.  Hamlet wonders what happens after death; the unknowable nature of the afterlife prevents him from killing Claudius when he catches him in prayer and stays his own hand when he contemplates suicide in the play's most famous "to be or not to be" soliloquy in Act Three.