Hello jhillman,
Welcome to the HP Forums! I understand your concerns when installing the Photosmart Premium C309G. How will you be connecting this printer? Wireless or USB? Please follow these troubleshooting steps to try and resolve your issue:
First, please follow this Microsoft document and run this tool on Fix problems that programs cannot be installed or uninstalled.
Next, perform a level three uninstall on your computer:
a) On your keyboard, press the Windows key and the letter "R" key to open the Run windows. In the search bar type %temp% and hit enter.
b) Locate the correct folder starting with "7z" (the correct folder contains "Util" and "CCC" folders).
c) Open folder "Util".
d) Open folder "CCC".
e) Please run the L3uninstall.bat.
f) When the uninstall has completed, restart the computer.
Before, installing the software please perform a "Clean Boot" on your Windows 8.1 computer. I will provide a Microsoft document on performing this step here.
Now, try installing the software from the HP website here. Make sure you choose your correct operating system; either 32 bit or 64 bit.
If the installation is complete, then please go back to the Microsoft document on "Clean Boot". Follow the steps where it states "How to reset the computer to start normally after clean boot troubleshooting".
Please inform me your troubleshooting steps.
Looking forward to hearing from you.