400- 450 dollar gaming pc

Try - CPU: AMD A10-7850K 3.7GHz Quad-Core Processor ($124.95 @ SuperBiiz)
Motherboard: Asus A88X-PRO ATX FM2+ Motherboard ($86.99 @ Newegg)
Memory: Apotop 8GB (2 x 4GB) DDR3-2133 Memory ($65.98 @ Newegg)
Storage: Seagate Barracuda 250GB 3.5" 7200RPM Internal Hard Drive ($48.25 @ Newegg)
Case: NZXT Source 210 (White) ATX Mid Tower Case ($38.99 @ SuperBiiz)
Power Supply: SeaSonic 300W 80+ Bronze Certified ATX Power Supply ($39.98 @ Newegg)
Total: $405.14
 
Without monitor, OS, peripherals and optical drive:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($108.99 @ NCIX US)
Motherboard: ASRock H81M-DGS R2.0 Micro ATX LGA1150 Motherboard ($47.99 @ SuperBiiz)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($49.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($51.49 @ OutletPC)
Video Card: Sapphire Radeon R7 260X 2GB Video Card ($115.98 @ Newegg)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($38.99 @ SuperBiiz)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($58.99 @ SuperBiiz)
Total: $472.42
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-06-03 07:50 EDT-0400

If you need to cut down the price, get a 500GB HDD instead of 1TB.
 


I plan to use a TV and Linux Ubuntu for now. I already have a mouse and keyboard.
 
If you go with the AMD APUs, like the one i7Baby suggests, you can then add a very cheap card later to utilize it's Dual Graphics capabilities and boost it's graphics considerably (be aware the card models that can do this are few in number so you'll need to research). Here's a good article giving an overview of that APU and how well it does Dual Graphics.

http://www.techspot.com/review/781-amd-a10-7850k-graphics-performance/page7.html
 
As far as gaming performance @ $450 is concerned, imho this is the best OP could find:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($108.99 @ NCIX US)
Motherboard: ASRock H97M Anniversary Micro ATX LGA1150 Motherboard ($64.89 @ OutletPC)
Memory: A-Data XPG V2 8GB (2 x 4GB) DDR3-1866 Memory ($47.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($48.75 @ OutletPC)
Video Card: Sapphire Radeon R9 270X 2GB Dual-X Video Card ($124.99 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($29.99 @ NCIX US)
Total: $448.59
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-06-03 08:37 EDT-0400


H97 motherboard will improve upgradability. No AMD APU could touch Radeon R9 270X performance.
Its like F1 vs Kart 😀


 


As far as gaming is concerned, Haswell i3 perform really fine because:

1. HT allow 4 threads.
2. Fist core do most of work anyway.

And Haswell Refresh i3-4160 has a really strong core.

Anyway as i said before H97 motherboard will improve upgradability so i5/i7 Broadwell is a real possibility for a cpu upgrade (2016/2017).
 


Absolutely.

I just implied that a Haswell I3 paired w/ a cheap H97 motherboard will give best bang for the OP buck as far as gaming is the main concern.

And could give OP a meaningful upgradeable platform.

http://www.xbitlabs.com/articles/cpu/display/core-i3-4340-4330-4130_5.html#sect0



 
I just wanted to inform you that AMD drivers for Linux are horrible, and would not recommend AMD Video Card for Linux user EVER if they want to do gaming with their PC