You will need to fish either a 14-2 or a 12-2 wire (the same as in the existing plug) to the new box in the bathroom. You can also determine the correct size from the circuit breaker rating. You will have to locate that particular breaker in any case; a 15 amp breaker can use 14-2 wire (or 12-2) but if it is a 20 amp breaker it will require 12-2 wire. If in doubt, use 12-2 (12 guage, 2 conductor plus ground).
Wires should be simply spliced into the same color wire in the existing box and terminated on the appropriate terminals on the new GFI. Make sure you terminate them on the "line" terminals, not the "load" terminals. Most new GFI plugs have a piece of tape across the "load" terminals but they are marked as well.
The only real problems you might encounter would be actually fishing the wire to the new location or if the existing plug is a GFI itself. If it is the new wire will need to be spliced into the existing wire with wire nuts rather than terminated on the same screws; you cannot put two wires onto one terminal on most GFI plugs.
You should be aware that the existing plug is probably on a 15 amp circuit breaker and this will limit the amount of current available in the bathroom. It will run a hair dryer, but probably not two of them at once.
If this does not answer your question, please either email me from the email listed on my profile or leave a comment on my hub about wiring plugs (http://hubpages.com/hub/how-to-install-electrical-... Either way I will get back to you.