Is this a good budget build for 1080p?

Here is a better one: http://pcpartpicker.com/p/XpHknQ
I wouldn't reccomend the power supply you chose as it isn't recomended for gaming. Look here for reference: http://www.tomshardware.com/forum/id-2547993/psu-tier-list.html
Your mobo also wasn't so good. Look here: http://www.tomshardware.com/forum/id-2384030/motherboard-tier-list-am3-chipsets.html
I also gave dual channel so it would be a little faster.
Do you already have a case and periphereals ( mouse, keyboard, monitor)?
Hope I helped.
 
This one will be a bit faster:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($70.99 @ NCIX US)
Motherboard: ASRock FM2A78M PRO3+ Micro ATX FM2+ Motherboard ($46.80 @ SuperBiiz)
Memory: G.Skill Sniper Series 8GB (2 x 4GB) DDR3-2400 Memory ($52.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($49.99 @ Newegg)
Video Card: XFX Radeon R9 280 3GB Black Edition Double Dissipation Video Card ($169.99 @ Newegg)
Case: Silverstone PS08B (Black) MicroATX Mid Tower Case ($31.50 @ SuperBiiz)
Power Supply: EVGA 600B 600W 80+ Bronze Certified ATX Power Supply ($51.30 @ SuperBiiz)
Total: $473.56
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-05-20 14:26 EDT-0400

The 860K is faster than the FX-6350 on gaming and the R9 280 that i chose has higher clock speed. BTW, i added a case, i don't know if you already own one.
 
Solution
PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-6300 3.5GHz 6-Core Processor ($85.50 @ SuperBiiz)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($31.99 @ Newegg)
Motherboard: MSI 970 GAMING ATX AM3+ Motherboard ($78.49 @ Newegg)
Memory: Team Zeus Yellow 8GB (2 x 4GB) DDR3-1600 Memory ($49.99 @ Newegg)
Storage: Hitachi Deskstar 1TB 3.5" 7200RPM Internal Hard Drive ($40.94 @ Amazon)
Video Card: XFX Radeon R9 280 3GB Black Edition Double Dissipation Video Card ($169.99 @ Newegg)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($58.99 @ SuperBiiz)
Total: $515.89
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-05-20 14:25 EDT-0400

- The FX 6300 is as good as 6350 when you overclock it to the same level.
- EVO cooler enabling a nice overclock
- Better motherboard
- cheaper ram and harddrive, without losing performance
- better PSU
 
I always suggest that people get an SSD for every build, no matter how small or how cheap, you can find one for like $50 these days so it will not break your budget.

You PC will boot up crazy fast, games start in an instant, the overall computing experience will improve. There's simply no better way to upgrade PC for non gaming purposes than an SSD.