HOW DO I CONNECT A COMPUTER RUNNING WINDOWS 8 TO A HOME GROUP RUNNING WINDOWS 7 HOME PREMIUM?

Note using Win Key + X and Win Key + W (to get to Control Panel, Run, and Settings as needed). Win Key + D calls the desktop and the using Win Key toggles Desktop and Start Screen.

As needed translate these to Windows 8 methods (there is little real difference) :
(The links below are marked Windows 8, Windows 7, or General)
All computers need to have unique names.
All computers need to have the same time and date.
Some antivirus/security programs can block or cause networking issues. Disabling a 3rd party antivirus/security program may not be sufficient. You may need to uninstall it and then run the maker's Removal Tool to prevent trouble causing remnants. Use MSE while testing or permanently.

Your router and all computers must be IPv6 capable (hardware, firmware, and drivers) to use HomeGroup. Check the System Maker (Motherboard Maker for custom systems) or the actual device makers' sites for update network drivers. Also check the router maker's site for updated firmware. Check their support and ask in their forums for any known issues. (Run the Home Group troubleshooter to enable IPv6.)