Probably the cheapest way to do it is to use a USB capture card. Personally I use the EzCap116 which works nicely.
You can hook the cable and device directly into your computer and play using capturing software. I recommend AmaRecTV, which will allow you to record video, play with aspect ratios, deinterlace, full screen and so on.
If you want to play on your TV while recording on your computer, you'll want to split the signal. I use a stackable composite splitter similar to the ones sold here on eBay:
If you want to see an example of this setup, here's a video of somebody connecting their Xbox 360 in this manner; it will work the same for the Wii.
EzCap also has their own official YouTube channel, with a setup video, which uses 3x female-to-female phono splitters. I recommend doing this so your video quality stays consistent between the displays you are splitting to.
A word of warning though: products named "EzCap" are made by an official company, products named "EasyCap" are bootlegs, as stated on their drivers FAQ.
You may also want to check out EzCap's official website.
Follow up: I have been using this setup for a while, mostly for Wii videos myself. I wanted to comment on the stackable composite splitter that I previously recommended.
The splitter I recommend above is not a standard cable. You will not find it anywhere in standard retail outlets -- only specialty shops and bootleg type cables.
The reason for that is because this specific cable suffers from a severe loss of quality on at least one end. The colors are dark and washed out. I do not recommend this cable if quality on both displays is something you are concerned with. If this is the case, go with something else, such as the separate splitters mentioned above, or get a switch box that handles this.
The phono splitters are also dirt cheap.
I regret buying the cable. I will say, however, that it is very convenient to only have one piece of hardware that splits your composite signal into two. But it is not worth the loss in quality nor the price.
Unfortunately there is no standard way of recording game footage from games consoles without extra hardware.
Some games have video capture tools built in (like Halo Reach), but if they allow access to those videos outside of the console ecosystem it can be in many different ways. For your Halo Reach example, you actually require an account on http://www.Bungie.net. Once this account is linked to your Xbox LIVE gamertag, you can access your fileshare from the Profile->File Share option on the website and access any videos you have saved.
As this functionality is Bungie / Halo Reach specific, this won't necessarily work the same for Modern Warfare 2 (or any other games). So this is where a specific piece of hardware; either a capture card that slots into your PC or an external capture unit such as this one that I know games testers working on real games use. Check out http://www.hauppauge.co.uk/site/products/data_hdpvr.html.
Obviously specialised hardware can have quite a high cost associated with it, but it does allow you to record every part of gameplay as you see fit and would work with any games console, not just Xbox 360.
Best Answer
Yes it is indeed possible, but not with the build-in recording tool. You would require a screencasting software that records your video output as well as your microphone input.
Here's a list of well-known screencasting software
GTA doesn't save the images your graphic card generates. It only saves something like a screen-play or a blueprint of the scene you're recording.
It saves movement, orientation and interactions of objects such as your character, NPCs, projectiles and more. For example when you record and fire a gun GTA will save your action (shoot) as well as your current coordinates and orientation (where you are & which direction you shoot). This way of recording saves a lot of resources and makes recording way less hardware-hungry.
Later when you watch this recorded movie it sets up the game the way it was when you were recording and then just moves all the recorded objects the way you recorded them.
I haven't found any resource that displays how it works but i looked at multiple documents and i saw that its really hardware efficient. I'm pretty sure that GTA uses this technique of recording even tho I don't have articles or files that approve this.