To add to this ,i dont think core 2 quad's have SSE4.2 instructions ,
so even upgrading to that or core 2 extreme for that matter will make the upgrade useless as majority or rather 95% of modern AAA games use SSE4.2
I dont know how cheap you can get the hardware in Philippines but according to my understanding you would profit more buying used parts if possible with some warranty if thats possible too in the budget and you can always stress test the parts to just to be sure
If its any of your interest ,i had built a cheap PC using used second hand parts for gaming for around 187 USD ,you could go lower perhaps there
As since you have core 2 duo ,your RAM is also DDR2 ,so for modern gaming you would need a new motherboard ,a new cpu ,new RAM and a new PSU (which make sure you get a new one and not used)
Edit - ah most importantly a Graphics Card too
If you need any guidance just hit anyone of us back with how much you're getting which part for and a better assessment can be done though its true in that budget ,used parts would suit you better albeit also testing them before getting them