If the machine and scren have VGA output and input use them. If not and they both have a composite video output/input then you can use that you simply need a phono cable to connect them. VGA is the better connection and gives much better resolution so use that if you can but composite out will work too.
As for powering it, you can either shell out about £100 for a good Opus 12v power supply made specifically for Carputers or you could try using an inverter... personally I'd got for the first option...
Its not a simple matter and therefore not cheap to convert vga to video.
PC monitors run at frequencies of 31Khz horizontal & at least 60Hz vertical (non interlaced as well) whereas Tvs require a 15Khz horizontal and 50Hz vertical, interlaced signal. The RGB & HV sync signals also needs combining into the single composite signal.
If your LCD accepts S-video input this will give a sharper image.
If you are using a PC, can you not fit a graphics card with a composite video output included - that's the cheapest option?
unless its a touchscreen then its not going to be that useful- you will still have to use a keyboard and mouse. Most of the non-touch screens only have s video or composite input, great for dvd headrest screens but not much use as a main control screen.
Inverters have there good and bad points too- its not just like a mains switch, and anyway, you do not specify what kind of computer you have- if its made to be put in a car then it may expect 12v anyway.
You really need to research this a bit more.... i suggest the forums at mp3car.com would be a good place to start.