In general, you want enough ram to have 1GB (for the operating system) plus 1GB per character being played.
If you don't have enough ram for this, it is the cheapest upgrade to improve performance.
Without enough ram, you might be able to launch the toons but performance will suck.
You'll need a 64-bit version of window, to be able to use more than 3.2GB (approximately) of total system ram (Ram + Video + Sound memory).

Once you have enough ram, the cheapest upgrade you can do is getting an SSD for the gaming folder.
Not sure if you have one of these or not, based on the limited specs you listed for your system.
For a gaming folder a 64GB lower end SSD is fine; you only need a higher end model if you're putting your OS on it too.

In general, 1GB of video ram (on DX9) is good to run 5 characters or 1.5GB (on DX11).
Again, not having enough video ram might not prevent you from playing, but it will negatively impact your play.
If you upgrade your video card, the double cards (two cards, on one physical card) generally suck for boxing (580 is better than a 590).
The best bang for your buck, is the GTX 670, but it isn't necessarily cheap.

What kind of an i7 do you have?
People have 10-boxed on an i7-920 in the past.
A generation 2 i7, such as an i7-2600K is a much stronger processor.