You probably CAN do it yourself, but if you're not familiar with the codes, I wouldn't. There are special codes that apply to bathroom outlets (need GFI, as you said), so either study up and do it ...

Adding an electrical outlet next to an existing light switch is easy, as long as there is a neutral wire in the box. Step 1: Turn the power supply off to the light switch at the main electrical panel.

Learn how to add an outlet above or below an existing one without the need for mudding, taping, or painting. This video demonstrates how to run electricity from an existing outlet to a new location ...

Expert DIYer April Wilkerson shows how to add a floor electrical outlet by running power underneath existing flooring.