Q: I have a dark grey screen when turning on my computer.

It would be helpful if you indicated the EXACT model MBP that you have and the current OSX.

Ciao.

Also, take a look at this Apple Support article for basic troubleshooting steps:

If your Mac won't turn on - Apple Support

On the back/bottom of the MBP is the serial number.  Enter it here and post back the model/year information:

https://checkcoverage.apple.com/

Ciao.

You have a late 15" 2013 MBP.  (you are advised not to post serial numbers of your MBP.  Please keep that in mind for the future )

Start with these basics, reset the SMC and the NVRAM:

https://support.apple.com/en-us/HT201295

https://support.apple.com/en-us/HT204063

Then try a safe boot:

https://support.apple.com/en-us/HT201262

If there is no change, boot into the recovery partition:

https://support.apple.com/en-us/HT201314

From the 4 option menu select Disk Utility.  Run Disk Utility>First Aid, Verify and Repair.

Post back your results.

Ciao.

So the SMC did noithing?

Did you try booting into the recovery partition?  (COMMAND + R)

If no success, the an appointment at an Apple store becomes the best option.  The evaluation will be FREE.

Ciao.