I know many people here will just automatically assume...but since there's no mention of it...
The choice of whether or not you should build a new PC, upgrade, or just stick with what you have, all depends on what you need it for. Why do you want a new PC? Why do you want to upgrade?
If it's for better gaming performance, then look into upgrading the graphics card first, and later on (after some more months of saving I guess) replace the CPU and motherboard with an Intel.
If you think the PC just feels slow, then first get a SSD (I'd say 120GB minimum these days for the OS) and if it's still slow, then upgrade the CPU.
If it's just for multimedia and homework and internet browsing, then you don't need to upgrade at all.
If you're going to use your new PC for gaming and give your old PC to someone else, then say so!
Also, nVidia GTX Titan Black is the most powerful single-GPU graphics card in the world. I forget if it's also more powerful than the current dual-GPU cards, but anyway. Costs around $1000.