An Intel Core 2 Exreme will take you from 2 cores 2 threads to 4 cores 4 threads, from 3.00 GHz to 3.2 GHz CPU, and from 1333 MHz to 1600 MHz bus.
Unfortunately it will also take you from 65 Watt to 136 Watt TDP.
If your PSU is as old as your current processor you will need a new one. It's likely that any kind of upgrade will require a PSU upgrade.
I suggest you start saving your money for a whole new system. It's hard to say what might blow next but with a 12 year old system, putting a new processor into it will certainly strain all of your existing components.
This is a matter of opinion, which you will get a lot of here. The core 2 duo generation of CPU and the underlying hardware running it is old enough that any upgrade you do to it is not going to get you much performance in AAA gaming titles today. You can probably find some parts on Ebay for next to nothing that will upgrade it a little bit but you would really be better served taking that money and buying any new computer. An entry level PC today will have significantly more performance than anything of that generation could deliver. A huge amount of change in the industry has occurred since the core 2 lineup that isn't just in the CPU and the ability to even run a current gen graphics card is questionable.
While you could upgrade to a Core 2 Quad or a Xeon, the system is so old, more recent games (think 2015 onward) won't run great on either as well. Even a cheap chip like a ryzen 3300X or i3-9100 will be noticably faster and draw way less power. So unless it's a personal project for fun and tinkering there is little reason to invest into it.
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