How do I get TV audio into my computer sound card with an HDMI cable?
HDMI cables have sound capability built in. The problem may be somewhere else.
A breakout cable will most likely NOT work! Most HDMI-compatible devices use copy protection systems which will prevent a straightforward solution from working. You may be able to plug the HDMI cable into your TV, and use the audio out (e.g. headphone socket) to connect to your PC.
Buy an HDMI breakout cable. It will have an HDMI plug on one end and separate plugs n the other for video and audio. You will probably still need to get an adapter to change the audio from L/R RCA plugs to a stereo 1/8 plug.
All of this though is dependant on your tv having an HDMI output. Most only have an input. If the tv has an RCA audio out plug then all you need is a regular stereo RCA cable and the 1/8 adapter mentioned above.
HDMI doesn't work that way, I'm almost certain. Your computer outputs, but doesn't accept a signal there. If you want audio from your TV to play through your computer, the line-in jack is a better bet.