What are the options of getting from Dresden (Germany) to Český Krumlov (Czech Republic)?

You can buy electronic tickets from Dresden to ?eský Krumlov from the Czech Railway's "eShop". The ticket will be sent to your e-mail address as a PDF file and costs CZK 1087 (appr. €40.20) for a full-price refundable ticket or CZK 662 (appr. €24.50) if you can make use of the "First Minute Europa" offer (must be bought at least 3 days in advance, limited number of tickets, non-refundable, bound to a specific connection). Currently, "First Minute Europa" tickets are still available for most connections on December 27th.

Travel time by train is between 5:45h and 6:00h and I sincerely doubt that you will find a faster connection using public transport. If you rent a car, Google Maps estimates 4 hours for the ride, but expect steep surcharges for one-way international rentals unless you are also going back to Dresden.

Trains are available, but neither the Czech Railway company, nor the German "Deutsche Bahn" are able to give you a price quote online. Both pages list reasonable train connections and show that you either have to (1) visit a ticket office, or (2) call to buy a ticket (or two). Note that calling does not mean that the ticket will be sent by post - rather you get a pickup code for the ticket vending machines at the train station in Dresden (or elsewhere).

There is a tool (whose link I cannot post) for finding price estimations for European Rail journeys. It seems to be 39,40 Euro one-way, but at German Railway counters, it may be more, as they use their own tariff for the German part of the journey. According to the EuroSpezial offer page, there may also be the possibility to get a saver's fare (which is cheaper but inflexible), but you cannot check this online as it seems.