I'm not sure if $500 is enough to get you a gaming PC good enough to run Bioshock infinite at high settings, although if you pick carefully you should be able to run it at low- medium settings with enough tweaking of the settings, I'll post some specs in a minute
EDIT: Okay, here's a fairly basic build that costs just over $500, and the price only includes the actual computer itself and not any keyboards, mice, or any other periphirals.
The build has a 3.1 Ghz 4- core processor, 4GB of 1600Mhz RAM, 500GB of 5400RPM HDD storage, an AMD Radeon R7 260X GPU and a 600W power supply. I didn't add in an optical drive because I rarely use them and it would just add to the cost of the build. Also, I'm not usually an AMD type of person as I normally go with intel and nvidia, but AMD is really the only option you have if you want to keep the price down or you're on a strict budget, so here you go:
http://pcpartpicker.com/p/2Vl2Q