To record all game sounds including chat including your own voice at the same time without having to use a separate recording device or microphone mixer, you will need an aftermarket gaming headset with microphone monitoring such as the Turtle Beach X31. This type of headset mixes together your voice from the microphone along with all the other game sounds including other peoples' voices in chat so that they can all be heard through the headset's speakers. We are going to be recording directly from the headset's speakers.
First, remove the batteries from the headset. Gently pull off the earcup padding surrounding the left speaker. This is the side with the microphone boom and audio controls. Remove the three screws surrounding the speaker to expose the circuit board. Be careful when removing the screws so that they don't get sucked into the speaker grill by the speaker's magnets. Also be careful because the speaker will still be attached to the circuit board by very thin wires.
Remove the two screws holding the circuit board to the plastic housing.
Solder three wires to a stereo 1/8" 3.5mm headphone jack from Radio Shack.
Remove the threaded retaining nut from the outside of the 3.5mm jack, drill a hole through the plastic housing of the headset and insert the 3.5mm jack through the hole.
Screw the retaining nut on from the outside to secure the jack.
Use heat shrink tubing or electrical tape to insulate any exposed wires on the jack to keep them from shorting out when everything is put back together.
Reattach the circuit board using the two screws.
Locate the left and right speaker solder joints on the circuit board. They are marked "SPL_L SPK_R" underneath the joints and "L+ L- R- R+" on top.
Solder the 3.5mm jack's right channel wire (red in this case) to the R+ joint, the left channel wire (white in this case) to the L+ joint and the common ground wire (black in this case) to both the middle R- and L- joints. These are very small solder joints so be careful not to apply too much heat.
Replace the speaker cover using the three screws and replace the earcup padding.
Plug one end of a male 1/8" 3.5mm stereo patch cable into the jack.
Plug the other end into the recording device's audio inputs.
You may have to change the wiring configuration because you are now recording directly from the headset's speakers. Run the Xbox's audio out only to the headset's RF input box and then the headset's new line out directly to the recording device's audio input. When you connect the audio passthough line out of the recording device to the TV, the volume of the TV will be affected by the headset's volume level. Go to the Xbox dashboard and then select Preferences and then Voice and then select either "Play Through Headset" or "Play Through Both". You may need to adjust the Voice volume and Game volume as well. Start a game or chat session and start recording. Adjust the headset's volume control until you have an acceptable recording level. Thanks for looking!