the 560 is not going to be able to play games at max settings to being with. But the i3 would be able to handle most games with no problem whatsoever.
Recording however is a whole other issue. Video compression is harder on a CPU than most games are, and running a game while doing real time high resolution video compression is quite a task which will require at least a quad core CPU to do it properly. If programs such as fraps started supporting quicksync then it would not be an issue, but as it is now, you really need to look at an i5 minimum.
As also mentioned by someone else, having a good storage subsystem is a must. Most HDDs will be able to handle the raw throughput, but they do not multitask very well, so you want to record to a drive that is not being used for your OS or games. RAID1 or 5 is suggested so that you have some amount of drive failure tolerance in your system, which is important when dealing with video because it is difficult to properly backup large files all the time.
A single large SSD has enough throughput, and low enough seek time to run the OS, game, and recording at the same time, but you will still want a 2nd drive to back up files with.
Lastly (as you seem to be on a budget), keep in mind that you do not 'need' a current system in order to max out games while recording. A slightly older non-K Sandy Bridge i5 with a decent quality H61/67 motherboard can game and record at the same time without issue. But then again, you get what you pay for, and what you want to do is going to demand some fairly powerful equipment.