Need ideas for gaming computer parts :D

So... here is the story. I currently have a 400$ Acer laptop that works, but it only gets me 20 fps in LoL on the lowest of settings, without any other tabs opened. I can't Skype with my friends, listen to music, and play at mid settings minimum. I have been searching forums and trying to come up with a build on this website - ca.pcpartpicker.com - and I have had no luck. If you have the time, and you know computer parts, please click or copy and paste that link and search around for some decent computer parts that work, get a good 60 fps, a case that will fit the parts, a monitor that is at least 60 hz, and fans to cool off the cpu and computer that will also fit that will make the whole computer round up to my budget that is 1200$ max. Thank you so so much if you do this, and I appreciate it beyond your imagination, THANKS!