I would recommend Elgato, specifically this: http://www.amazon.com/gp/aw/d/B00840353W/ref=redir_mdp_mobile If your really on a budget, iMovie should cover your needs, but if you are looking for slightly more serious option, Final Cut Express 4 would be great for your situation.
What is on your TV that you want to capture? Is it something you can get online?
To capture an input that is going into your TV, you need a capture card or device. For HD capture devices, you're looking at about $150-$300 for the device. Something like This capture device.
If you're platform gaming I know Elgato makes an adapter that allows you to play to your TV and record to your computer at the same time in full 1080P HD. If you can get recorded video, you can get recorded audio.
The basic Game Capture HD card works just fine on USB 2.0, and it's the only one of the models, to my knowledge, that can record retro systems, the other models: HD60, HD60 S, and HD 60 Pro, only take HDMI inputs. The only downside is that the standard HD card can't capture HDMI footage at 1080p 60fps.
As far as your processor power goes, you should just have trouble streaming video, not capturing. Just to be safe though, make sure that absolutely nothing else is running on your computer while you're working. I hope this helps!
https://www.amazon.com/dp/B00840353W/ref=twister_B00NTQ2JL0?_encoding=UTF8&psc=1