MBP turns off during boot

Question

Hello!
My mid-2010 MBP is having some issues. It is no longer under applecare. Last year, when it was still under applecare, I had a problem of the display going black many times per day while doing even simple tasks. I believe they replaced the logic board and everything worked just fine after that.
However, last night my laptop suddenly went black while viewing a PDF. I tried opening and closing the cover, nothing could get the screen back. At that time, the keyboard was still lit up and the caps lock button still worked (light would turn on and off when pressing).
I did a hard-reset.
The laptop screen lit up like it was starting, and then died. Tried again, nothing. So, I waited about five minutes and tried again. The screen lit up, the apple appeared, and then the laptop died.
After many attempts, I have seen it do many different things. Sometimes it dies before the apple appears. Sometimes it dies after that, but before login screen. Sometimes it dies on the login screen. Usually the longer I wait, the further it will go - however, even when stone cold and not warm at all after waiting hours, it will still only run for ten minutes before it turns off. Sometimes if I can get all the way through startup, it will run for a minute, or five minutes, or ten minutes, but it always fails. Sometimes it turns completely off, but sometimes only the display goes black and the keyboard stays lit. I can even change the keyboard illumination up or down when this happens. Its about a 50/50 chance whether the display will die or the whole laptop will die. Whichever one it is, though, it still requires 10-15 minutes before I can try again, or the machine will just fail halfway through startup.
I have also tried starting into bootcamp. It worked for five minutes in Windows and then died.
I have tried booting up while holding shift, but it was taking a very long time for the progress bar at the bottom to fill, and halfway through the laptop shut down.
As I said at the beginning, the logic board is one year old.

Answer 1

After sitting for a day, I started up the laptop and made it all the way through login. I let the machine sit without touching anything for about twenty minutes, nothing happened.
I then opened Chrome, and as soon as I began typing into the search box, the entire computer shut down, and restarted by itself.
Halfway through restart, the screen turned black. I waited, and after a minute the keyboard lit up so I knew it was at the login screen, though I couldn't see anything. I typed the first letter of my name, pressed enter, typed my password, and pressed enter again.
I presume the computer is logged in and sitting at the desktop now...I have the laptop set up so that if I move the cursor to one corner, it puts the display to sleep. When I do this with the track pad, I can see the keyboard lights dim, then light back up again when I move the cursor back out of the corner.
Anyone have any suggestions? How can I get my screen back?