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Premiere Elements will probably run OK on that computer. Download the trial and see if it works OK for you,
Premiere Pro CS4 or earlier may or may not, depending upon what media and effects used.
Adding more RAM and a better graphics card might allow you to use it on CS5 or CS6, but performance won't be very good.
Looking at your configuration (that's now 10 years old at this point, not including the GPU), 8GB is the maximum amount of RAM that your system (or motherboard, in particular) supports. The G41 chipset does not support 8GB DIMMs at all (or more specifically, the maximum amount of RAM per memory rank that the chipset and all other LGA 775 chipsets officially support is 2GB - this means a maximum of 4GB per double-ranked DIMM), and is limited to four ranks total of memory.
This, coupled by the fact that your PC is running on a CPU that's extremely sluggish by current standards, and the rest of the components in your system that are severely outdated (for example, that GeForce GT 420 is of the old Fermi generation that's proven to choke Premiere Pro in H.264 encoding, has only 48 CUDA cores and is running on horrible-performing DDR3 VRAM that's only 128 bits wide and with a throughput of an equally abysmal 28 GB/second), and I'd recommend not running anything media-related on that PC at all.
As such, you'd be WAY better off saving up for an entirely new, up-to-date PC build than spending a single penny on video editing software that requires newer and more powerful-performing hardware as time progresses.