A gaming PC could be a costly affair. Although your budget seems decent,if you want to save a little, you could try this:
Intel i5 4th gen
Nvidia GTX 750Ti
Kingston Hyper X RAM - 8GB total
The rest is upto you. I have that configuration and games like Watchdogs, Crysis 3, Skyrim , Assassin's Creed: Black Flag etc. work on ultra settings without lag.