If you set up with a DVD recorder so that the HDMI cable runs into that and then out again to the TV, you should be able to record your footage that way. Depending on what recorder you get, you could either burn it to a DVD or take the footage straight off the recorder's hard drive using a PC.
For the voice over, it wouldn't be easy to have both the game noise and the voice over recorded onto the same file at once, but if you don't care about the game noise then you should be able to set up the DVD recorder so that it takes input sound for the HDMI cable from a different input, such as coaxial or optical. If you could get a microphone connected to a coaxial or optical output, you should be able to do the voice over that way.
If you care about the game noise, the only thing I can think of is recording them on separate devices and just overlaying the talking later during editing (which you are bound to have to do anyway.)
