As Jim noted the PC uses multiple voltages both positive and negative so it is not as simple as bypassing the power supply, you need multiple supplies and different voltages, IT can be done, by I've never researched where to go to find them.
An easier way would be to use a laptop.. Many laptops run on a single DC supply and for around $100 more or less you can get a 12 volt supply for a laptop. Many laptops will work while closed, using an external keyboard, mouse and monitor.
To use the back-up monitor (Please do not call it a dash TV) as a computer monitor you will also need, in most cases, a device called a "Scan converter" and you may be limited to VGA video levels
Myself.. I use a laptop with S&T and use the back up monitor as a "Rear view mirror" save when I'm pulling into a gas station, then it's my "Pump view mirror" so I can line up on the pump.. I have plans for additional cameras to feed it.