By far the best value for your money will be getting a used Q9450 and a new Radeon 7850 for your existing build. Put the excess savings aside and slowly save up for a full build.
A Q9450 + 7850 will play any game with fairly good performance with the correct settings. It is around the same performance as a Phenom II at similar clock speeds. You can overclock even on a stock cooler into the low 3Ghz range easily, just bump up the FSB and check your ram multiplier to make sure you're not running too aggressive with memory clocks (google overclocking guides on Core 2 Quads, or start a thread here about it).
An AMD FX CPU is even better, and of course an Intel i5 or i7 2xxx/3xxx/4xxx is better still, but those would involve new board, new ram, new windows installation, reinstall of all drivers/apps/etc. The cost and time involved are serious, and your budget is not enough to allow for :
New board
New CPU
New ram
New GPU
*potential new OS
In your region. Prices for computer components altogether are considerably more expensive in Europe (hello taxes!), bringing your potential builds for 300 Euros down into the range where it makes more sense to just build on what you have within sane limits. I recently upgraded a friend to a Q9550 and GTX 660ti, with 6GB DDR2-800, he's able to play Battlefield 3, Deus Ex Human Revolutions, Crysis 3, Bioshock Infinite, etc, at very respectable smoothness at 1920x1080. This is on a P35 motherboard from ~5 years ago.
Once you are able to save enough money to do it right, a baseline will look something like this :
Asrock Z87 motherboard
Intel i5-4670K or 4770K CPU
Cooler Master 212 Evo Heatsink/Fan or better (Noctua DH14!)
8GB or 16GB DDR3-1600/1866
Radeon 7950 3GB or Nvidia 760
550W well-reviewed power supply
If you check the prices in France for something even in the range of a 7850, it is quite high :
http://www.pc-look.com/boutik/Prod_...-GDDR5-PCI-Express-3.0-x16-...__75647_en.html
Best wishes!