There are several ways to reach Elsinore from Copenhagen. An easy and beautiful one is the train, known locally as "Kystbanen" - "The Coast Line".
The train is departing from Copenhagen Central Station, Nørreport, or Østerport, the three biggest train stations in Copenhagen.
The train to and from Elsinore departs three times an hour and takes around 40 minutes.
You can also go directly from Copenhagen Airport, and that will take approximately 60 minutes.
Tickets can be bought at the station.
Elsinore can easily be reached via highway E47, and will take you 40-60 minutes depending on traffic.
You can also take the more scenic route along the coast. It will probably double the travel time, but in exchange you will get a really enjoyable scenic drive.
In the summer you can even go by Ferry. Three times a day Sundbusserne will depart and take you directly to the Elsinore harbour. The trip takes 1 hour and 20 minutes.