On one side of my wall (in kitchen, not the refrigerator or stove plug) is a three prong grounded outlet. Would it work without overloading a circuit to put a new outlet on the other side of the wall (the living room)? Any special precautions I should take? The house is small and short on plugs, so I could use a few more. Is this OK or not OK?
National Electrical Code says no.
210.52(B)(1) basically says, that all the receptacles in the kitchen should be supplied by 2 or more 20A circuits. 210.52(B)(2) says that there should be no other outlets on these 2 or more circuits, which means no lighting outlets and no outlets outside of the kitchen. 210.52(B)(3) says that the 2 or more circuits, can be used to supply receptacles in other 'related' rooms (pantry, breakfast room, dining room, or similar area).
Even though the kitchen and living room may share a wall; according to NEC 2008, they cannot share an electrical circuit.
Your question does not provide enough information for a definitive answer, but load calculations are not difficult.
First, you should only load a circuit to 80% of capacity, so a 15amp circuit should only have 12amps of load.
Amps = Watts/Volts. Add up the wattage of all the devices plugged into that circuit and divide by your household voltage. If you are in North America, this is 120V. Lets say you have ten 10 50watt lights:
Amps = (10 * 50) / 120 = 4.12Amps. In this case, you can safely add another outlet because there are about 8amps of capacity left. If this were above 10 I'd say you should consider adding a new circuit since you would be just on the cusp of overloading the circuit.
This answer only applies if your jurisdiction accepts the NEC as the model code. Even if it does not apply, it gives you some idea where an authoritative code organization stands on this issue.
You may NOT extend a kitchen outlet circuit to serve a living room. Generally, you cannot extend such circuits in a kitchen at all, however there is an exception for certain related rooms such as dining or breakfast rooms or pantries where it is acceptable to extend kitchen circuits.
As for determining whether an added receptacle would overload the circuit, as mentioned, the total load cannot exceed 80% of the circuit ampacity. How to determine this load has been misrepresented as it relates to NEC requirements. First add all permanently connected loads on the circuit, fridges, lights, dishwashers, disposers, etc. Then for all convenience receptacles not having a permanent device connected, assume each receptacle strap in a residential occupancy would draw 180va for load calculation purposes. The code recognizes some outlets are not used and others used could draw much more than 180va, some simultaneous, others possibly not. 180 is simply a happy medium to limit the number of outlets on any given circuit. One is not required to anticipate the actual load of every device connected to any particular outlet if it can be easily unplugged and moved elsewhere.