Usually if the games start lagging at higher resolutions, it is a sign the card is weak. The 5770 doesn't have quite the same memory bandwidth as some of the higher cards and may be bottlenecking your system. Keep in mind, though, your CPU is close to five or six years old at this point and would choke under some games like GTA 4 that prefer to have a couple more cores.
GTA 4, Skyrim, Rage (if I remember correctly) and a few others that slip my mind will utilize quad core.
Added note: Upgrading the CPU and GPU might also give a performance boost in games that don't use four cores. Improved processor architecture can make a marked difference. The i5 and i7 series at the same frequency (3 GHZ) of your CPU would blow past it due to other improvements, and those can be regularly overclocked to 4 - 4.5 GHZ.
I have an old build that's based on a Core 2 at similar settings with a better GPU (GTX 470). It handles most games reasonably well and lags out on GTA 4 pretty bad. I still use it from time to time, particularly if I have company over and we want to play a LAN game.
Primarily? I test by observation and playing with the settings. If I decrease settings that stress the GPU more than the CPU (resolution, shadows, textures, aliasing, etc) and get a large performance boost, that indicates most of the time a GPU limitation. If my FPS stays about the same no matter what resolution or settings I play at, then usually it's a CPU limitation. Long loading times are a sign of low hard drive bandwidth. RAM speed doesn't give the same jump in performance it once did due to virtually everything in the system being faster than a hard drive these days.
If you wanted to keep it cheap, you could look on Ebay for a Q6600 or a QX6700 (which is the unlocked quad core version of your chip). I've seen the Q6600 for under $100 USD and the QX6700 for around $130 USD.