There seems to be a significantly newer driver available on Logitech's
C615 support page. I'd start with this, after checking if the web cam is the standard recording device (see Dark123's post above).
If neither solves the problem, try without any drivers installed (and without other sound-related devices plugged in). To remove the driver, do this:
- Press the Windows key.
- Open the Control Panel.
- Go to "Device Manager".
- Navigate to "Sound, video, game controllers".
- Double-click on the C615 entry.
- Switch to the tab that says "Drivers".
- Push the "Uninstall" button.
- Remove the cam and restart your computer.
- Plug it back in (nothing else) and see if the mic works.
If it does, yay! If it doesn't, leave it plugged in and install the newest driver (even if it didn't work before).
Good luck!