a $600 PC will be better served as an Athlon II X3/X4 or Phenom II X4 than by an i3.
without knowing the exact prices in poland, something like this should work:
770, 785, 870, or 880 chipset
Athlon II X3 445/450, X4 640, or Phenom II x4 955
4 GB DDR3 1333 or 1600 MHz RAM
500GB to 1 TB HDD
a HD5770 or GTX 460 at either 768MB or 1 GB or a HD5850
high quality PSU (look at the Xigmatek in Today's article on Toms)
and an inexpensive case and DVD burner (Haf 912, Coomermaster elite 3xx series, NZXT, Antec 200 or 300, etc)
That PSU is quite good but a bit weak for a modern more powerful video card. You can either take your chances with your PSU and get a HD 5850 (better and consumes less than a GTX 460) but you might still have to get a new PSU. Theoretically it will work but not much spare room left.
The problem is that more and more games are very CPU dependent and they don't work well on dual cores like the i-3. Getting that CPU would be a mistake, better get the Athlon II X3/X4 or Phenom II X4 (but your PSU can't take that 125 watt CPU).
About the PSU, it's the 12V rail wattage that's important, you only have 28A (336 watts). A HD 5850 needs some 150 watts on full load (a GTX 460/768 even more), that Athlon II X4 CPU - 95 watts, and there's the HDD, DVD-RW and the case fans also on the 12V rail.
If you still want Intel prepare to increase the budget by a lot, you need the i5-750, an expensive SLI mobo and a new PSU. For budget systems AMD offers more than Intel.