Most of the advice in this thread is applicable to residential home construction, not RVs. Some of it is scary.
Twist-on wire nuts are not appropriate in an RV. Crimp connectors are. Receptacles with wire-wrapped screws are not appropriate in an RV. Positive locking wire connections are.
(For reference, notice that ALL automotive connections are crimped or positively locked. There are no wire nuts, twisted wires, screw connections, or even soldered connections. And that's all 12v, not 110v.)
The way a professional would do this job is to open the self-contained outlet that has power and add a power-tap that connects the new self-contained outlet to the existing line. That might mean some additional cable in either the existing circuit or the new circuit to obtain the needed slack.
Of course, if the existing circuit is already fully loaded, such as for an air conditioner or microwave, then a different circuit will have to be found.
How it should be done is described in this installation guide (p. 10): http://www.hubbell-wirecon.com/literature/InstallationGuide.pdf
Also note that whenever you mention self-contained devices used in RVs, someone will post that they are cheap and dangerous. That is not true. They are more expensive than residential devices and you can find UL and CSA certified devices such as the Hubbell Wirecon devices. They are used by RV manufacturers since they are thin and light. Like their residential counterparts, there are good and bad ones. Choose wisely.