Sigmund Freud, the inventor of psychoanalysis, was born in 1856, almost two and a half centuries after the death of Shakespeare in 1616. Thus there is really no way that the play Hamlet could have been influenced by psychoanalysis. While one can apply psychoanalytic theory to almost anything, as it purports to give a universal account of human nature, it would not be wise to do so for two reasons, first that its application to Shakespeare is anachronistic and second that traditional psychoanalytic theory has, to a large degree, been rendered obsolete by more modern, scientifically based accounts of how the human brain functions.
Although there are many different ways to stage Hamlet, I would choose an attempt to reproduce many elements of its original staging, albeit with female actors playing female roles. The reason for this is that many of the plot elements of the play, including Hamlet's outrage over his mother's marriage to Claudius, are clearly grounded in religious and court traditions of his period. Also, it just strikes me as glaringly incongruous to have people speaking Elizabethan English in productions set in other periods (although more abstract settings can be effective).
The main thing the creative team would need to do to stage such a production would be careful research to guarantee authenticity in everything from costumes to movement style and props to accents.