In the US, we're allowed to have GFCI protected regular outlets in our bathrooms. I understand this is not the case in the UK.
Anyway back to the original question: First of all wihello, if you are not experienced with electrical work you probably shouldn't be starting with bathroom wiring. There are extra regulations to follow and working with electricity in a potentially wet environment is more dangerous.
First I'd take a look at that existing outlet. Is it grounded? Is it GFCI protected? Is it installed with the proper polarity? If the answer to any of these questions is no, or you do not understand any of them, you definitely want to get an electrician to do this for you.
Before you even consider doing any work, you want to have the proper tools on hand. Get a voltage tester and a screwdriver that's insulated all the way up the shaft, up to but not including the tip. You'll also want wire stripers and a wire cutter, and 12AWG wire. If you don't want to go digging around inside the wall, you'll probably want to get a surface-mount outlet and wiremold so you can run everything on the outside of the wall.
Overwhelmed yet? Get an electrician to do it for you. It's a simple job and won't be all that expensive. Otherwise, continue:
Get yourself a do it yourself book and follow the instructions carefully. Don't skip any steps. The wiremold will have instructions on how to install it on the package. Make sure you have a location picked out before you start drilling holes, and remember the carpenter's rule (measure twice, cut once). Use a level to get attractive, straight lines. Mark things out with a pencil first as a guide.
Most importantly flip the circuit breaker off before you touch anything! Double check that the circuit is dead with your voltage tester. An amateur should never work on a hot circuit. Work safe, electricity is invisible and it can kill you if you're not careful. And since this is a bathroom, make sure the floor is clean and dry. Before you screw on the covers and flip the power back on, use the voltage tester in resistance or continuity mode to test for short circuits.
Also, don't work alone. If something does happen, someone should be around to help you.