Recording with high quality and high bitrate has almost no impact on the CPU and can be done with almost every CPU out there. And yes, APUs and cheap stuff like a mobile i3 work too.
First: you wanna use AMD VCE, NVenc or QuickSync to remove all the load from the underpowered CPU. Set the bitrate as high as you need for that, use variable bitrate (constant bitrate is needed for streaming, not for recording).
Second: your hardware is not capable of displaying any normal games with decent settings at 1080p with 60FPS even without recording, so you better forget that now and do some shopping.
What you need for 1080p 60FPS streaming: decent overclocked i7, decent GPU to get 60 FPS min. in games that you actually wanna stream.
For recording: decent i5 or AMD FX will do really good, i3 and APU work just fine but require more tweaking. Again, you need a GPU that gives you enough FPS and still get enough power to record at the same time.
My suggestion for a decent rig that can keep up with everything that you can throw at it:
i5 6600k, Z170 board, 8GB DDR4, GTX 960 4GB or R9 380X minimum. Better GPU will give you better FPS and higher settings without dropping FPS. Better CPU (i7/Xeon) will give you better quality on stream/recording, again a bit more FPS in some games and it will help with other work like editing&rendering videos, multitasking and is just generally worth it for a content creator.
