I have had good luck with low-cost Dell Inspiron Desktops, to which I add a better power supply ($50 from Newegg, and takes about 5 minutes to install) and a video card of your choosing.
An Inspiron with an Intel i5-6400, Windows 10, 8 GB RAM, 1 TB hard disk, DVD+/-RW, keyboard, and mouse is about $550, so this route might be too expensive. If you go with an i3-6100 instead, it is $450.
So, total then would be $450 (computer) + $50 (PSU) + $200 (graphics card) = $700. Still a tad over budget.
However, I have an older, slower Dell than that with a GTX 750 Ti card (cheaper than the GTX 960), and it plays AH3 at about 60 fps with the slider set to "none" and other things set to default.