Please try this and report back if it worked or not.
Reboot, run Skype. When it shows you the blank blue background as shown in your screenshot, leave it open. Now open the Run dialog (WIN + R) and type or copy & paste the following:
On a Windows x64 system:
On a Windows x86 system:
"C:\Program Files (x86)\Skype\Phone\Skype.exe" /secondary
Press Enter or click OK:
"C:\Program Files\Skype\Phone\Skype.exe" /secondary
This opens a second Skype process in a new window. If this second instance does not work either, it tells something is wrong with your Skype installation and you should completely remove it from your system and reinstall. If the second instance works as it should we need to look for the error somewhere else.