There is a simple solution for this, Windows has build in microphone devices that allows you to record what Windows is playing. This feature is however, tricky to activate it. To activate, press right click the sound icon inside your system tray, press Recording devices. A window will show all installed recording devices, then press right click at the device list and select "Show disabled devices".
If you're have up to date sound drivers, there will be a device called "Stereo mix", press right on it and click enable. Now you can select this device at the microphone selection of your favorite chat program.
For more information about this topic (in case updating the driver won't help you), see: