Yes there is a systematic way to trouble-shoot. Just treat this as a new build and follow the checklist. Here it is: http://www.tomshardware.com/forum/261145-31-perform-steps-posting-boot-video-problems
I really think that the motherboard needs replacement, and this would be the place to start.
However, if too many major parts need replacement, start thinking of a new build. Salvage whatever components you can from the fried computer.