If the screen physically fits in the lid on top of your dash you can put it there. How to mount it, is something that is up to your imagination. (browse around the "Show your project" forum a bit to get some idea's.)
Most people use systems based on Via Epia Motherboards, these are socalled Mini-ITX motherboards, they measure 17x17cm and have most things onboard. (video, lan, audio, usb2 etc.) There also are Nano-ITX boards, 12x12cm, but be prepared to pay alot for those since they're quite expensive. You can also use a normal desktop PC, but keep in mind that a desktop system consumes more power.
A keyboard is not required, but this is depending on the software you are going to use. Most frontends use on screen keyboards, but it's still handy to have a normal keyboard in the car.
You can use the factory car stereo. There are 2 ways to to connect your pc to the stereo:
1. Using the Aux in line of the car stereo. (alot of older stereo's do not have this option though.)
2. Using a FM-Modulator. You place this between the antenna and the antenna-in on your stereo. You can select a frequency to broadcast on and tune to that frequency on the radio to hear the sounds from your pc.
Hope that helps.