Create me a Streaming PC Build

Hey Everyone! I'm currently trying to go from playing league on a really, like REALLY crappy laptop to hopefully a good Desktop PC that can be able to play league at 60fps or higher at 1080p on (High settings?) and hope to stream with too. Let me just show you what i'm dealing with right now with my current laptop specs.

Laptop: HP 2000 Notebook (LOL, Got it back in 2013, started playing league early 2015) SPECS: CPU: AMD E-300 APU with Radeon(tm) HD Graphics

GPU: AMD Radeon HD 6310 Graphics

RAM 4.0 GB

Operating System Microsoft Windows 10, 64-bit (Originally Windows 8).

So yeah, it sucks LOL. Usually get 15-25FPS(All graphics as low as possible) during laning phase. During late game team fights, forget about. Lucky if i stay above 5fps.

This would be a huge upgrade for me. I would also hope to stream on this new rig because i've always liked to entertain and it would be fun. So i would like you guys to create me a part-list cause i'm not so knowledgeable when it comes to picking out pc parts.( https://pcpartpicker.com/list/ ) Here are some info though,

Budget: $550-$650 (Can go a little higher) Operating System: Windows 10 Home

My friend also created a part-list for me to, i just don't know if its reliable: http://pcpartpicker.com/list/mbW98K You can use that as a starting point i guess.

Thanks!