Ok - here is the best build that you can get, with a monitor, for less than $800.
PCPartPicker part list / Price breakdown by merchant
If you are primarily gaming, it is better to have fewer but stronger CPU cores/threads rather than more, weaker cores/threads thus, the i3 is a better option, for you, than the 6300.
I've gone for a nicer case because that Azza would be pretty nasty tbh.
I've also got an SSD in there - it's definitely worth it, just get one, you won't regret it.
I've also gone for a nicer monitor and a semi modular PSU.