This is ENTIRELY possible, although you would need to either buy or build some hardware to make it happen. In addition, the software solution would be very, very complex.
The computer's power supply would be out of the picture for the most part. You need to read data straight from the wall with as little extraneous noise as possible. From the electrical engineering perspective, this is a very thoroughly covered topic. In the end, all you're really doing is an analog to digital conversion, and the rest keeps your circuit from being fried.
The software solution would basically be eliminating random noise, and looking for embedded signals. The math behind analog signal analysis is very complex, and you can spend a few semesters in college covering the topic, and the rest of your career trying to master it. If you're good at it, there's a cushy job for you on wallstreet predicting the stock market.
And that only covers reading incoming signals. Transmitting is a whole 'nother sport.
Now, it also sounds like you might be interested in a hack. That is...
You could buy a
commercial-off-the-shelf power-line
Ethernet adapter and tear it apart.
They have two prongs that plug into
a standard wall outlet. You could
remove these and wire them to the
INSIDE of a power supply.
To do that, you'd have to tear apart a power
supply as well, which is incredibly
dangerous and I hereby warn you and
anyone else to NEVER attempt this.
The entire Ethernet adapter could be
tucked into the power supply and you
could basically have an Ethernet
port on the surface of your power
supply (either inside or outside the
computer).
Simply wire that to a
standard Ethernet adapter and voila
(!), you have nothing but a power
cable connecting your computer to
the wall outlet, AND you magically have
Ethernet!
Note that there also has to be another power-line
Ethernet adapter somewhere else for
you to establish a network and make the whole project useful.