There are a lot of portable batteries around, designed for using externally with laptops, which can charge by being plugged into mains power (and have a nice convenient power cable coming out).
With regards to the power consumption, you're probably looking at (approximately):
CPU: 25W
Video: 20W
Board: 25W
RAM: 5W
HDD: 5W
DVD: 5W
Keyboard & Mouse: Insignificant
Total: 85W
This is in the unrealistic situation that all components are receiving full power draw at the same time.
Adding a screen will push that up further, but if you consider the ~35W figure for a 17" LCD to be a maximum, you're still only talking about 120W=75% loading. Still OK for a 160W (if you go the 120W route you won't be powering the monitor as well).
As for the monitor, if it does have a standard RCA port on it (make sure of it before buying), then conceptually it could work. You may need to plug into an external monitor while installing drivers, etc... Keep in mind that the image quality will be drastically reduced if using RCA.
There are also portable LCD monitors available, some of which have their own battery packs. A laptop monitor wouldn't work easily (it's nowhere near as simple as plugging it in).