if your upgrading the processor to then an core i5 is all you need for gaming. so I would get one of the newer ones if you can afford it, for cpu bound games and future.http://www.newegg.com/Product/Product.aspx?Item=N82E16819117372&leaderboard=1
as for the graphics card it is somewhat of a personal choice between NVidia and amd Radeon. so you can check out the article linked above or check out something in the NVidia 760 range or Radeon 270.
If he/she didn't want to upgrade the CPU, they wouldn't really have to. The i3-2130 in that Acer Veriton already performs better than an FX-6300 in some games, and never performs too much worse.
A GTX 750 TI is probably all I'd risk on the power supply and in a case with so little room, so a GTX 750 TI and replacing the RAM with 8GB would probably offer the most practical benefit for the price, and would turn it into a competent mid-range gaming PC, comparable in performance to the Xbox One.