$450 budget pc (max league of legend settings) w/ or w/o monitor

Solution
If you don't care about upgradability then this is a good PC:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($68.99 @ NCIX US)
Motherboard: Gigabyte GA-F2A78M-D3H Micro ATX FM2+ Motherboard ($57.99 @ SuperBiiz)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($59.98 @ OutletPC)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($48.89 @ OutletPC)
Video Card: Gigabyte Radeon R7 360 2GB Video Card ($109.99 @ Amazon)
Case: Cooler Master N200 MicroATX Mid Tower Case ($38.99 @ NCIX US)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($57.99 @ SuperBiiz)
Total: $442.82
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-07-18 11:23 EDT-0400
 
Solution

delaro

Judicious
Ambassador
PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($71.98 @ Newegg)
Motherboard: Gigabyte GA-B85M-D3H Micro ATX LGA1150 Motherboard ($69.99 @ Directron)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($54.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($48.89 @ OutletPC)
Video Card: Gigabyte Radeon R7 360 2GB Video Card ($109.99 @ Amazon)
Case: Cooler Master N200 MicroATX Mid Tower Case ($38.99 @ NCIX US)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($57.99 @ SuperBiiz)
Total: $444.82

LoL doesn't require much hardware, this will give you the same FPS or better and the ability to upgrade. The FM2 builds are what they are with no upgrade future left, I only consider them when I can pick up the parts at a ridicules price.