Lightroom Classic does not have a serial number. After you install LrC it will find the catalog you have been using with Lightroom 5 and will prompt you to upgrade it. You should allow this to happen because it is necessary. The upgrade process will create a completely new copy of the catalog. That new copy will have the same name as the old one, but will have a 10 appended to the name. It will be the catalog that LrC will use, and should start right where you were working previously.
LrC functions solely using your Adobe ID and password, verifying that your membership is paid current. I leave the creative cloud application manager active and remain signed in to it at all times. Personally, I think that is the best and most reliable way to work with LrC.
If you install Lr Classic, you shouldn't need to install and open LR 5 on your new computer. It will open the catalog file (and convert it to LrC).
Also, for future reference, you need to keep track of your serial numbers (of all software, not just Lightroom) better ... don't let it happen that you can't find serial numbers. I keep them in a text file which I store in "the cloud".