Usually for game capture computers use hardware like capture cards. A solution would be to use software instead. However that puts a heavy load on the CPU and without a beefy CPU, the computer will receive a huge performance decrease. I use a program called Bandicam. There's a free version and a paid version. Both work just fine. The reason I use Bandicam is it has hardware acceleration which means the CPU intensive video encoding is moved to the GPU where it can do it faster and more effecient.
Basicly Bandicam with hardware acceleration = similar performance of a $100 capture card.
However Bandicam doesn't capture Xbox or PS3/PS4 video. You need a capture card to do that.
That's enough information about what your son is interested in. Now for the computer advice.
Since you don't know much about computers as you said then building a computer yourself isn't the best option. The computer you listed is ok for what your son wants to do but the item description has left out what video card the computer has. That ibuypower may or may not include a video card. In the gaming world the video card is the most important part of the PC.
A good option instead is to buy a PC with the AMD A10 series processors. They're quad core (same as the FX4300) but they have the video card chip inside the CPU. The A10 series is very good for people who are on a budget but still want decent gaming performance. When your son gets older and out grows the power of the A10's video chip you can simply buy a new video card and monitor for a couple $100 and still keep the A10. (the new A10's support Bandicam's hardware acceleration)
I bought the A10-5800k a year and a half ago for $120, motherboard, ram, and powersupply. The total cost came to about $400. Those are just the basic components for the computer. You will of course need speakers, keyboard, mouse, and monitor. You can get a small cheap monitor for $50-$100. Most prebuilt computers will come with a keyboard and mouse. Since your son wants to do video recording a $20 or $30 headset will do. I'm not sure you're comfortable with your son using a webcam but that's also another option.
A 1TB hard drive is probably overkill for a 12 year old. I have a 1TB hard drive with about 30 or so games installed and I've got about 400gb of space left.
Forgot to mention. The computer will need some way to connect to the internet. Most prebuilt computers don't come with WiFi cards. If you have a long enough ethernet/LAN cable you can connect it to the computer and router.
Below I included a link to a custom built computer. If you want to take the time to learn to build a PC there are plenty of videos out there that can show you how a computer is put together. Many people think its hard to do or the parts are fragile, it isn't. Just follow the steps and computer parts now days have fail safes. Putting a computer together is like those playing with those toddler toys that teaches about shapes. Put the triangle in the triangle slot, square in the square slot. It's almost impossible to plug a cable in the wrong spot.
Link dump time.
About the A10 CPUs. Scroll down and click on "Specs". You can see a list of all the A series CPUs and compare them. You want the ones in first table. AMD's "APUs" gain considerable performance increases with faster RAM.
http://www.amd.com/en-us/products/processors/desktop/a-series-apu#
A suggested computer.
http://www.walmart.com/ip/Acer-Aspire-ATC-120-UR11-Desktop-PC-AMD-A10-7800-Quad-Core-3.50GHz-8GB-DDR3-Memory-1TB-HDD-DVDRW-Windows-8.1-64-bi/40900204
Same computer, but cheaper.
http://www.newegg.com/Product/Product.aspx?Item=N82E16883101017
It's out of stock and its less powerful than the Aspire ATC above.
www.newegg.com/Product/Product.aspx?Item=N82E16883281171
I thought I could build a PC for under $500. Didn't quite make it. Included a monitor and WiFi card. $564-127=$437
Close to the Aspire ATC but with better quality parts. I left out the keyboard, mouse, and speakers since those can be bought for cheap at walmart or other computer store.
http://pcpartpicker.com/p/BTFB4D
A list of PC related videos to help with building or buying a PC that I've collected. You don't need to watch every one, but I recommend watching the first 3, the ones about RAM speed, and "how to build a PC" parts 1 and 2 as they show step by step installation of 2 computers.
http://www.youtube.com/playlist?list=PLR_YoBC188kCPD2BglvfErscd1QvRQxDX
If you have any other questions let us know.
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