I have faced a Black screen problem recently, I tried everything but I could not fix it , so I decided to reinstall the operating system , but the thing is that all my personal data were stored in the same partition alongside the OS, so I booted from a Live USB and deleted all the windows files (boot,win,sys32..) manually, except my personal data , the question is how can i install a new operating system without losing my files ?
Ps: I don't have an external hard drive to back up my files.
Assuming you're on Windows 10, it's not possible since Windows 10 is touted to be a secure OS. All the data that was on it is wiped. You could use Recuva or TestDisk but results cannot be guaranteed. To answer your question, it's not possible the OS will need a formatted partition before installation can begin.
If it's a prior revision of Microsoft's OS, you can plug the drive to another system and retrieve the data from the partition but then again you're going to need to know what you're looking to retrieve.