Hi,

It seems to be an issue with the drivers not loading properly. Please try to download and install the following drivers:
Click this link to install the System Software: http://dell.to/RR9x8L. 
Follow these instructions to download and install the drivers:

• Click on 'Download File' and click 'Continue'.
• Click 'Yes' on User Account control if prompted.
• Click 'Continue' on the Application prompt and click 'OK' on Select the folder where you want to unzip the files to (make a note of the folder location).
• Click 'Yes' on the next prompt for folder does not exist. 
• Click 'OK' on All files were successfully unzipped.
• If the installation wizard does not appear, go to the folder location, select the Setup file; right-click and select 'Run as Administrator'.
• Follow the installation wizard instructions. Reboot PC.

Once done, please update BIOS from this link: http://dell.to/RziJN4. Follow the aforementioned instructions for downloading and installing the drivers.
Before flashing the BIOS please check the following:
* No external devices(except the keyboard, mouse and monitor) should be connected.
* All other programs should be closed and documents saved.
* The AC adapter should be connected during this process. The battery should be 10% or more than 10% charged.
The laptop will reboot after this update.
Once done, click this link: http://dell.to/T8lBQ4. Click 'Video'>> select the driver depending upon the video card (integrated or discrete) installed on the computer. Follow the aforementioned instructions for downloading and installing the drivers. Reboot PC.

When the computer is performing normally, push the Display button (no.3 in the image) on and off and check if the screen turns off on on.

Hope this helps.
Keep me posted with the status. 

Unfortunately that didn't solve my problem. I did everything you said.

With the button I can turn on and off the display but only if it was on. But when I do a cold start it doesn`t work. I mean the light goes on/off when I click it but the display stays black.

I'm not even sure it's a driver problem since I can't even see the Dell booting logo at the Bios level where the Windows drivers not even apply. I think there must be something else wrong.

Update:

I found something out. When I leave the laptop open and do a cold start it starts normally. But when I close it and then later open and start it the black screen problem occurs.

So it must have something to with the closing of the laptop. Maybe that is helpful in figuring out what the problem is.

Thank you for the response.
Please share the following information:

• Where was the hard drive replaced from?
• What is the power plan that has been configured on the laptop? Click Start button >> type power options in the Search Box and press Enter. Click 'Change when the computer sleeps' and click 'Restore default settings for this plan' and observe the computer with the lid closed.
• Please share the advanced settings that are selected for the power plan (refer to the image) before restoring the defaults for the power plan. Click 'Change advanced power settings' and expand the list for 'Sleep', 'USB settings', 'Power buttons and lid', 'Display'; share the options that are selected.
  
  
  
   
  
  
  
• If you tilt the screen to a particular angle, does the display go out?

I have added you as a friend. Please accept my friend request by clicking on my name highlighted in blue and then click on “Friends” tab at the top and then click on “Request to Review” and finally click on “Accept” button. 
I am sending you a private message as well. Click on “Inbox” to respond to the message and provide system’s Service Tag and contact details so I may access your system records and check for further course of action. You could also click on Start Conversation to send a private message.

Keep me posted with the status.