Hello thnorris, and welcome to Eight Forums.
No. You will not need to purchase a new Windows 8 or 8.1 product key. Since your Dell laptop came preinstalled with Windows 8, it's product key is embedded in the UEFI/BIOS firmware chip, and was entered automatically when you installed Windows 8.1. You should be able to update to the 8.1 RTM (final) the same way but you will only be able to keep your personal files.
If upgrading from Windows 8.1 Preview/Windows RT 8.1 Preview to Windows 8.1 final/Windows 8.1 RT final (when released October 18th 2013), you will "Keep personal files only", but will lose your settings and have to reinstall all of your apps.
Hope this help,