PSU: http://www.newegg.com/Product/Product.aspx?Item=N82E16817139049 GPU: http://www.newegg.com/Product/Product.aspx?Item=N82E16814125491 While looking over for the specs for the PSU, I noticed it only had 1 PCI-E power connector, while the graphics card takes two. Am I simply misreading this or do I need a different PSU?
The R9 270 uses 150W of power. I don't know what other equipment you have in your computer, but it doesn't sound like you will have sufficient power. I recommend at least a 500W PSU for an otherwise standard PC build.
If you're not sure how much your computer consumes, try to virtually build the system on http://pcpartpicker.com (including all disks, fans, optical drives, CPU, RAM and everything else that consumes power). It will let you know if you have any problems with your build.
On the Gigabyte page they state that that GPU requires a 500W PSU with two 6-pin external power connectors. The PSU you link to is only 430W and only has one 6-pin external power connector.
Look for PSU with at least 500W and with two 6-pin external power connectors