For recording and streaming 1080p content from PS4, XB1, WiiU or Xb360, I personally use an Elgato HD60. It's a bit old by now, and there are probably newer, better versions available. It was about 150 bucks when I got it. This box does not have analogue input, so if you plan on streaming or recording from older consoles (or PS3), this would not be the device for you. However, older devices can usually be emulated on PC, in which case you could stream it directly from your system without any additional equipment.

For your PC setup, you should ideally have at least a dual core i5 at 2 ghz at the minimum. If you want to multitask, or have an overlay of your own face or something on the stream, you should probably have a 2ghz quad core at even higher clock speeds.

For software, I use OBS Studio, because it's free and does the job. It lets you add multiple video and audio sources, and also has a chroma key (if you want to do something with a green screen) plugin if you want something like that. The interface isn't the best, however.

I usually stream to beam.pro, because of the very low latency from gameplay to online stream. Twitch usually has a 15 second or higher lag, while I typically see less than 5 seconds on beam.