Some .mkv videos I try to play using Media Player Classic won't play audio. Those same files will play audio when played with VLC. DVD's also play without audio. This leads me to believe there could be a hardware issue between the motherboard and the audio card. However, I do not know the googlefu terms to research this possibility myself. Does anyone have any input to offer?
It's much, much more likely that the videos in question have audio encoded in a format MPC doesn't recognize but VLC does... If some files play fine on both, then it's not a hardware problem.
Find two files, one with audio on both players, one with VLC only. Open each in VLC and look at the "Stream Information" window for the audio stream codec... There is where the difference exists.
This is surely a codec problem.
Both MPC and VLC were originally developed by Gabest. However, MPC is now a community project, and its currently active fork is Media Player Classic Home Cinema.
VLC comes with its own codecs, while MPC uses the DirectShow codecs that are installed in the computer (same as used by Windows Media Player). Apparently you are lacking some codecs.
To install a very complete codecs-suite, download and install the K-Lite Codec Pack.