Your best bet is by taxi from in front of the station. I can't remember exactly how far it is but I would be surprised if it costs more than $20.00 - which will be alot easier than taking a bus or the tube.
If you don't have much stuff get the Victoria line light blue tube line to Green Park and either walk from there or change to the Piccadilly line to Hyde Park Corner. If its like my last visit there, be prepared for a long wait.