My Sony VAIO Laptop screen is black. It worked fine, no issues2 days ago. Went to turn it on last night and it is black.. When light hits at a certain angle, I can see the desktop, I can hear the startup sound etc, dvd door is working. I would like to make repairs myself if possible. Where do I start looking.

A faint and barely visible image on the screen indicates that the backlight(s) that illuminates the display isn't powering on. A device called an inverter board takes DC power from the laptop motherboard and converts it to AC power to power the screen backlight.
Before attempting anything below, be sure and remove the laptop battery and unplug the laptop
You need to take the screen apart and check to make sure that no cables are loose that are connected to the inverter board. If everything is secure, the next thing I would do is try to test the inverter board by hooking up an LCD from another laptop to the inverter and turning the laptop on to see if it lights up. If the other screen does light up, then the inverter board is likely good and your backlight needs to be replaced. Backlight replacement, unfortunately, can be very tricky for even an experienced technician, and many will opt to just replace the entire screen.
If you can't test the inverter board, then it might be worth it to order one and see if installing it fixes the problem.
Youtube is a great place to check for instructional videos on laptop repair, such as this one:
http://www.youtube.com/watch?v=jjXLR5yY7X4
Good luck!