About why Shakespeare would have set his play at a castle based on Kronborg, in addition to what's mentioned on the Main Page, there is also surviving documentation that a company of English actors played at Helsingor in 1585.  Presumably, that would have been part of the celebration of the Castle's completion.  Further, actors who later became part of Shakespeare’s company are known to have given performances at the Danish Court.  Then, English actors played in Denmark, at Copenhagen, for the coronation of King Christian IV, in 1596.  Thus, there was significant travel of actors from England to Denmark in that era.  Shakespeare had persons in his own profession to talk to, about Kronborg Castle, and Denmark.

About why Shakespeare would have set his play at a castle based on Kronborg, in addition to what's mentioned on the Main Page, there is also surviving documentation that a company of English actors played at Helsingor in 1585.  Presumably, that would have been part of the celebration of the Castle's completion.  Further, actors who later became part of Shakespeare’s company are known to have given performances at the Danish Court.  Then, English actors played in Denmark, at Copenhagen, for the coronation of King Christian IV, in 1596.  Thus, there was significant travel of actors from England to Denmark in that era.  Shakespeare had persons in his own profession to talk to, about Kronborg Castle, and Denmark.