iTunes Library - Help with transition to new computer - Backup Catch 22 Situation:
Old version of iTunes
Old version of Windows Operating System.

Cannot backup library because I cannot access my account without updating iTunes and iTunes will not update until I update my Windows operating system.

Is there a way to backup the library to an external disk drive, purchase a new computer and restore from that drive?

Alternate:  How do I get iTunes to backup my existing library?
There is a third party solution to this and it's freeware. But what it does is backup your entire user profile and then restore it on another PC.

I have been using the ForensIT TransWiz to migrate people from one computer to another. I did a whole bunch of Windows 7 to Window 10 migrations and as long as all you need is located in your user profile folder (iTunes library is) this works a treat.

Basically you log onto your PC as an admin user but not as the person you want to copy. Run the program and save the created file to an external drive. When you have the new PC log on as an Admin user (don't create the user you intend to copy to) and run TransWiz again, point it to your previously backed up file and it will restore to the new PC.

Once you have done all of that install a new copy of iTunes and it should find your old iTunes files and update them for the new version.