As far as bang for your buck goes, an Xbox One would be better.
Consoles are like that because everything between them is standardized and produced in bulk, whereas an equivalent PC will cost you ~$800-1000.
A decent gaming computer will cost at least $600-$700, assuming you have a monitor, speakers, etc. Right now the PCs I have seen that surpass consoles run for 1000-2000, depending on how much overkill you want.
To give you an idea of it all, my computer cost me $450, with the monitor, speakers, keyboards, etc adding another $275, and I can play new releases on high/medium. Focus on a great GPU and CPU, the rest should revolve around that.
Well depending on how much you get for it, it's possible. However, unless you already have peripherals such as a monitor etc and a copy of Windows 7/8, a lot of your budget would be going into that. Although, if you were to save some money up, or have some money saved up already, you could build a pc for about $750 that would completely and absolutely blow the Xbox out of the water.
It's up to you really.
Edit* I only just loaded up the other posts, which I completely agree with. But there's my two cents.