Modding the PSU isn't the way to go about it, as the mains power (be it 110, 220, 240, whatever, AC) is always-on.
Now if you would like the monitor to have a hard power cut when you turn the computer off that is quite doable.
Grab a relay with contacts rated for at least 5 amps AC and a coil rated for 12v.
Run the monitor power from the outside world into the case, through said relay, and to the monitor.
Take a 12V output from the PSU and wire that to power the relay coil.
Turn computer on, 12V turns on relay, turns on mains to monitor. Presto!
Now, the disclaimer:
Messing with mains AC is a great way to get yourself killed and start fires. If you aren't familiar with it, relays, AC safety, and electrical wiring in general you really really shouldn't be messing with it.
Bonus disclaimer: Mains AC is safer that the primary side of the PSU, it's even easier to kill yourself with if you open it too soon and touch the wrong spots, or if the bleed resistor is dead and you touch the wrong spots.
Now that we're past the disclaimer:
You could crack the PSU open and source the mains AC from the inner side of the PSU power plug. This would reduce cables at the cost of opening the PSU (not the safest thing) and doing even more messing with mains AC.
EDIT:
Easiest way to do what you would would be to get an LCD aimed at automotive use. They're 12v.