The main reason to go to Helsingor, other than to catch a ferry to Sweden, is to visit Kronborg Castle. The outside of the castle is the most impressive part, esp. when you arrive by boat from Sweden and can see it from the water. There's really not much to see inside, a few displays, but really pretty stark. It's probably more representative of what most castles were like back in the day - dark and dreary.
I arrived just before they closed, but the girl at the ticket counter let me in for a quick tour anyway. This is the castle from "Hamlet". Shakespeare never came here, but there's a plaque dedicated to him in the courtyard.