You need to start from the power source and work forward. First thing is seeing if you get a light on the Magsafe adapter. Next would be the DC-IN port. Use a multimeter and start checking.

Sounds like your logic board is bad or something is wrong with your battery that is killing it on start up. The Mac has four steps during start up, the first one being hardware check. If it fails at this point, your computer will still continue to run--you just don't see any flashing screen or chime. The issue is most definitely hardware related. Assuming your laptop is out of warranty, try removing all accessible hardware and starting up the computer. If that still fails, I would call it an issue with the logic board(assuming the battery is functional).