a couple of things you can try.
1. remove battery, remove hdd (usually can be removed from the bottom) remove 1 of 2 memory (bottom may show only 1, which means the other is on the other side, so take that one out).
2. try getting the dust out by blowing compressed air into side vents out the bottom and/or other openings.
3. plug in ac adapter, and try to turn it on. See if the shutdown issue goes away or not. make sure to NOT touch anything to make sure bad ac adaptor connection isn't causing shutdown issue (very common).
4. if still no display, try to plug it into external monitor (vga, hdmi, etc), and see if signal goes to external display (check manual to see how to toggle display, to make sure). if display comes on, you have lcd monitor or display cabling issue.
4. if still no display, you have no boot or no display failure (mobo, cpu, gpu, or ram)