The bin file is not supposed to be opened as it is the actual 'firmware' you are trying to upgrade to. Once you plug your WGA600N into your computers lan port you can use the built-in software to configure/upgrade everything you need. I'm not sure what software is missing (router/WGA600N or both) but once you plug the WGA into the LAN open up a browser & type 192.168.1.250 in the browsers address field (No www or http or anything else). You are now at the setup screen for the WGA & can select any setup you want. If you want to upgrade the firmware you can do that from the Administration page. From that page select upgrade firmware & select a path to where you have the .BIN file stored. That will take care of the firmware upgrade. You failed to mention what kind of router you are using so I'm assuming that too is Linksys.
Here are some links at Linksys forums where people who are having similar problems can help you better than here.
Linksys main fact/download page:
Linksys.com - Support / Technical Support / Choose A Product / Specialty Products / Game Adapters / WGA600N/Downloads
Forum for discussing the WGA600N including DTVDVR users.
Game Adapters - Linksys Community Forums