HDMI is kind of a bad choice because of HDCP, the encryption which is enabled when people watch movies for example. You're not guaranteed to have unencrypted output from the PC with digital outputs. Same for DVI and Displayport, these also have optional encryption. There's a chance you'd get encrypted output and it would be hard to decrypt it without paying a lot of money to get a key
You could use a FPGA to grab video frames from VGA output and send them to a hardware encoder chip and then mux the video frames and audio to a stream and send it. If you want to avoid licensing, I guess you could do motion jpeg, theora etc but you'll complicate your life.
There are such devices already if you're not aware, they're called KVM over IP .. they're expensive though.