I have really being wondering how I do this because I am trying to make a half decent halo reach recruitment video so I can actually get some clan members because I am rubbish at recruiting on matchmaking. Do I have to go to a specific website or what?
Right now the only way is using a capture card or getting a friend / finding someone with a card that can record for you. If you have an Xbox One though, Reach will be available as a backwards compatible title soon, once it drops you can then use the X1’s game dvr to make clips. You can edit on the X1 using upload studio, or upload your clips to OneDrive and then edit them on PC later 
Ouch. Been hoping to view a couple of my old vids, guess I am outta luck.
Yeah. But you can only view the player in first person if it’s the character the person recorded. So if you recorded yourself fighting another guy, you can only see the other person in 3rd person which stinks 
Capture card, either external or internal.
HDPVR 2 or Roxio are good ones to use. Just make sure you have the money to buy them and a good computing system that can handle it