Between Holland and Denmark lies Germany. You can look at the international site from the Dutch railways (NS) to find out whats the best travel option for you and what the prices are.

https://treintickets.nsinternational.nl/...

Here' s an excample

https://treintickets.nsinternational.nl/...

If you don't mind what kind of transport you take but you want to get there cheap you could consider to take a international bus, or look for a cheap plain ticket. The hitchhikers site is also an option.

International bus like Euroline. But look at their site to make sure were they are leaving from and what the destination is, if its outside a city you need to travel further by local transport.

http://www.eurolines.nl/en/

Cheap flight tickets

for excample

http://www.cheaptickets.nl/en/

hitchhiking

http://www.hitchhikers.org/index.cgi?lang=en

http://boeking.cheaptickets.nl/en/checkout/flightresults?cid=5931e15b-2da7-487c-9780-ddbf05e539f2&out0_dep=AMS&out0_date=20140912&out0_arr=CPH&cls=Y&adt=1&chd=0&inf=0#cid=5931e15b-2da7-487c-9780-ddbf05e539f2&out0_dep=AMS&out0_arr=CPH&out0_date=20140912&adt=1&chd=0&inf=0&cls=Y&pagesize=10

There are trains, but no direct trains. You'd have to change at least once.

There is the direct night train from Copenhagen Denmark to Amsterdam the Netherlands (also called Holland), once per night.

During the day you can also make the travel but not without changes.

If you are on the mainland rather than one of the islands, there are one or more stops on the night train within Denmark.

Yes, there are many trains between Denmark and the Netherlands every day.