How do I know if this video card is compatible with my computer?

It wont fit that motherboard is crazy small:
Form factor: uATX - 24.4 cm (9.6 inches) x 24.4 cm (9.6 inches)

The pci-e slot is way to close to the ram as well...

You may need to get an itx version of a gpu e.g...

https://pcpartpicker.com/part/msi-video-card-r9270xgaming2gitx

That SHOULD fit wouldn't worry about overheating be more concerned with psu wattage it is only 300watts which is not enough to run a 270x.

You would be better of spending $500-600 for a budget gaming build just my 2cents.

I would go with a 3258k as it inexpensive and allows a lot of overclocking headroom(Can overclock this cpu to around an i7 performance level in games with a good enough motherboard and cooler).

For gaming you would want to focus on the cpu and gpu as they are the most important things when building a gaming pc.
An example budget build from tomshardware that was popular.
http://www.tomshardware.com/reviews/build-budget-gaming-pc,4021.html


A build that I have posted before: http://www.tomshardware.com/answers/id-2451718/build-bf4.html

I think it is an improvement over the one above (faster gpu)

[PCPartPicker part list](http://pcpartpicker.com/p/mZKgK8) / [Price breakdown by merchant](http://pcpartpicker.com/p/mZKgK8/by_merchant/)

Type|Item|Price
:----|:----|:----
**CPU** | [Intel Pentium G3258 3.2GHz Dual-Core Processor](http://pcpartpicker.com/part/intel-cpu-bx80646g3258) | $64.98 @ OutletPC
**Motherboard** | [MSI H81M-E33 Micro ATX LGA1150 Motherboard](http://pcpartpicker.com/part/msi-motherboard-h81me33) | $45.99 @ SuperBiiz
**Memory** | [G.Skill Value Series 8GB (1 x 8GB) DDR3-1333 Memory](http://pcpartpicker.com/part/gskill-memory-f310600cl9s8...) | $67.98 @ Newegg
**Storage** | [Western Digital RE3 1TB 3.5" 7200RPM Internal Hard Drive](http://pcpartpicker.com/part/western-digital-internal-h...) | $49.50 @ Amazon
**Video Card** | [HIS Radeon R9 285 2GB ICEQ X² Video Card](http://pcpartpicker.com/part/his-video-card-h285qmb2gd) | $189.99 @ Newegg
**Case** | [Fractal Design Core 2500 ATX Full Tower Case](http://pcpartpicker.com/part/fractal-design-case-fdcaco...) | $39.99 @ NCIX US
**Power Supply** | [Corsair CSM 450W 80+ Gold Certified Semi-Modular ATX Power Supply](http://pcpartpicker.com/part/corsair-power-supply-cs450...) | $26.99 @ Newegg
**Optical Drive** | [Samsung SH-224DB/BEBE DVD/CD Writer](http://pcpartpicker.com/part/samsung-optical-drive-sh22...) | $13.99 @ Newegg
| | **Total**
| Prices include shipping, taxes, and discounts when available | $499.41
| Generated by PCPartPicker 2014-12-29 19:53 EST-0500 |

Some things to keep in mind this pc gets its performance from overclocking the cpu for gaming I would consider this a must so you will also need to buy a decent cooler such as a closed loop corsair h80i

http://pcpartpicker.com/part/corsair-cpu-cooler-h80i

If you dont intend on overclocking I would try this build:
http://elitegamingcomputers.com/good-cheap-gaming-computers/#21

EDIT: I may have cut things a bit close with the 450w corsair psu I would try this one instead https://pcpartpicker.com/part/corsair-power-supply-cs750m a bit more expensive but not cutting things so close this wattage