$1300 is a very good budget, but important questions:
Do you need an OS?
Do you need a monitor?
What else do you need that you want to be in that $1300?
What other tasks are you going to be using this for besides gaming?
Are you interested in overclocking your CPU?
Here's my suggested list:
http://pcpartpicker.com/p/LhZP
This build focuses on using quality parts that can be used in multiple builds, and has the tools to be very good at almost any PC job. It assumes you have your own peripherals (monitor, keyboard, etc.) and are not interested in overclocking, but leaves it open should you want to try it in the future.
As for going wireless, thats easy. You can buy a $20-$40 USB stick and instantly turn any PC into a wireless PC. Or you could pick out an internal card, but that isn't my area expertise.