Here's a full Mini ITX build with Monitor and peripherals for under $800:
http://pcpartpicker.com/p/rgffsY
It's not perfect, but the CPU is clocked higher (and still has 4 threads), more RAM, SSD, and a far superior GPU.
i3 -6100 3.7GHz HT Dual Core - Great cheap gaming CPU. Does not OC
ASRock H110m-ITZ/ac - Mini ITX 1151 motherboard with WiFi AC. Does not OC
2x8GB G Skill Ripjaw V - Cheap, and plenty of RAM
Samsung 850 Evo 250GB - OS + LoL installed here will be much quicker than the laptop
Caviar Blue 1TB - Very cheap and reliable storage drive
Sapphire R9 380 - Great 1080p card, arguably the best card for under $200 new
Thermaltake Core V1 - Tiny case that still looks cool
Corsair CX430 - People will say that there are better PSUs, and they're right. But this one is Semi-Modular, affordable, and decent enough for your budget
Sceptre 24" 60Hz 1080p Monitor - Higher resolution and better refresh rate than the laptops you've linked
Cooler Master Devastator - Cheap and cool looking peripherals