Good choice of components!

Ignore the GHz thing. the CPU boosts up to 3.6 using turbo boost, and it runs very fast. Note that there are more physical cores than the i3, and the performance is the next level up. Purely clock speed doesn't determine the power of a processor just for future reference.
That build looks great as is, but i'd go for 16GB of ram, much better long term choice as 16GB is becoming the recommended spec in recent times, and you will no doubt upgrade eventually.
If you want to cut down to your budget again, you can change out a few things. See build for reference.
The Seasonic is a great unit, but so is the EVGA 750w B2. It's made by Super Flower, which is a great OEM and currently has a $20 discount from NCIX with a mail in rebate, which leaves room for upgrades in the future, definitely worth it.
This is the least you can do with an i5. the 6400 is just a slightly lesser clocked version of the 6500, and is not worth it imo, but trying to stick in budget range here.
It's the best you're going to do if you want an i5 system close to your budget, although I recommend the next build, see below.
PCPartPicker part list:
http://pcpartpicker.com/list/YsHGcc
Price breakdown by merchant: http://pcpartpicker.com/list/YsHGcc/by_merchant/
CPU: Intel Core i5-6400 2.7GHz Quad-Core Processor ($179.89 @ OutletPC)
Motherboard: Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard ($53.88 @ OutletPC)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($66.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($47.49 @ OutletPC)
Video Card: PowerColor Radeon RX 480 4GB Video Card ($200.00)
Case: Deepcool TESSERACT WH ATX Mid Tower Case ($35.99 @ SuperBiiz)
Power Supply: EVGA 750W 80+ Bronze Certified Semi-Modular ATX Power Supply ($58.99 @ NCIX US)
Total: $643.23
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-07-26 05:23 EDT-0400
If you can go a bit higher, this is probably the best value build you're gonna get.
Try to convince your parents to spend that little bit more if that's the case, or save up for another week.
PCPartPicker part list:
http://pcpartpicker.com/list/nQzPnn
Price breakdown by merchant: http://pcpartpicker.com/list/nQzPnn/by_merchant/
CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor ($199.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B150M-DS3H Micro ATX LGA1151 Motherboard ($65.38 @ Newegg)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($66.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($47.49 @ OutletPC)
Video Card: PowerColor Radeon RX 480 4GB Video Card ($200.00)
Case: Corsair SPEC-01 RED ATX Mid Tower Case ($49.95 @ NCIX US)
Power Supply: EVGA 750W 80+ Bronze Certified Semi-Modular ATX Power Supply ($58.99 @ NCIX US)
Total: $688.79
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-07-26 05:26 EDT-0400