Make sure that you are working safely by testing that the outlet is dead by either plugging a radio or lamp into it or using a voltmeter or non-contact voltage detector. Tape the breaker off so that no one else turns it back on while you're working.
Try not to buy very shallow boxes in either case. You could even add in a 4 gang box (with 4 receptacles giving 8 plug-ins) if there is room between the studs in the wall.
(Most outlets are single, though, meaning they have just two places to plug into.) The new box can be any size you want. If the existing box has two outlets in it (four places to plug into), a "two-gang" box will be necessary. One of the old work boxes should match the existing box. Two new plastic "old work" or "cut in" boxes (see above photo).Single gang box (left) two-gang box (right). Although not strictly necessary, it is a wonderful addition to your toolbox and can protect you from shock hazards. A non-contact voltage detector is always very nice to have. You will be working in either an attic or crawl space, and you will need some light to work by. A pair of needle-nose pliers will be very handy to have. You will need a Phillips-head screwdriver, and a flat blade will be handy as well. You need a way to measure, because locating the place to cut into the wall or the hole to be drilled must be done with precision. Electrician's diagonal wire cutters will be useful. Wire strippers are preferred, but other tools will do. You will need a way to cut wire and strip insulation off of it. It will be very handy to have a metal-cutting blade for the jigsaw, or some other saw blade that will cut nails-even a plain hacksaw blade will do. You could use a razor knife (box cutter), a jab saw for drywall, or a jigsaw for paneling (or drywall). A small square will need to be cut into the wall where the new outlet is to be installed. A cordless drill is ideal, but if yours has a cord, you will need an extension cord as well. A second, smaller bit around 1/8" can be very useful. A drill bit of about ¾" will do the trick. Some form of drill will be necessary, one that's capable of drilling through about five inches of wood.