In the novel, Whitby is where Dracula came ashore in the form of a dog after the boat he was travelling on sunk. The church there is where some of the action took place. That apart it is an attractive place to visit in it's own right. The Magpie Cafe in town is reckoned to be one of the best fish and chip places in the country!
Can you get a cheap flight to Newcastle (Aer Lingus and Ryanair fly there from Dublin), at least you will end up on right side of the country. Pick up a car and it is about a two hour drive to Whitby from there. Depending on the time of day you arrive, you could have a quick visit/have a lunch break in, to the World Heritage city of Durham on the way.
Use this route planner to see distances involved and logistics of the various travel options
http://www.theaa.com/route-planner/index.jsp#
Regarding using Ryanair - be very careful you add all the extras to the headline flight cost. Very often when you factor in their very strict and low baggage allowances, fares from other airlines look more attractive!
If I were you I would probably go on an organised day trip to Salisbury and Stonehenge from London.