(Yet another) Upgrade PC or Buy a new one

If i were you i would buy a new one. Just keep the computer case and hard drives.

i5 650 seems to be dual-core. Amount of memory is likely also bit too low...

Is that some sort ready built computer? If so I would look going to fresh "custom", depending on skill level either self-build or build by store.

You can try to update your video card. In the case, you can use it for the new pc.

The core i5 650 is comparable to a low end Core i3 (based on interpolation from Anandtech's Bench), and these are typically decent enough for most games. I believe that a good GPU would be enough to get your frame rates into the 30+ rate with high settings. That said as has been mentioned 3GB of RAM is a bit low.

If all you want is, as you say, to be able to run the latest games with "decent graphics quality with 30-120 fps" (at 1080p, I assume), then you can buy something like a GeForce 660 (or even 650 Ti Boost) or Radeon R9 270, throw in 8GB of RAM, and you will have a gaming rig for a £200 upgrade which will be able to play pretty much all games at very high quality level (though a few might not play at the very highest level).

You can of course go for a higher end card, which will support all games at the highest quality level and be more future proof, but a £200 upgrade should be make your gaming life significantly better.