Yea you could. I went the route of buying used parts. You can usually get them for half off or less on most parts by getting used parts. So you essentially get twice as much with your money by buying used, and even sometimes refurbed components. I usually try to get parts that have not been rigorously overclocked or worked real hard. It really is a great way to get nicer components when on a budget.
I don't know what country you are located in. But this forum has a deals section that you can find nice parts for good prices usually. Often times the parts are performance oriented, and from my experience, the people I have dealt with have been very kind and honest. Also, I have found some pretty nice deals on Craigslist. But that depends largely on the are you live in.
Now you can definitely reuse your ram and HDD. Remember that your GPU is the main factor in gaming performance. Followed by CPU and ram. But you already have 8 gigs of 1600, so your fine there.
Honestly...This is what I would do in your situation. Using the methods described above, I would get a high end GPU and power supply first. Next I would upgrade the motherboard and CPU...which I would not go with anything less than an FX-6300. An Intel i5 4690K would be ideal, but would cost more...about £280 with the motherboard. Then last, I'd get the SSD, case, and whatever else you might like to upgrade or acquire. And who knows. Maybe you can find everything together used for a sweet deal somewhere. But by upgrading in that order, you would be getting the better performance soonest.
But that CPU is not horrible. And who knows, maybe you would even be happy with it when paired with a high end graphics card. But it could definitely be better also. It all depends on how nice you want your games to look, and how much you are willing to spend to get there. But you really only need a higher end GPU, CPU, and motherboard to get high or ultra settings on most games.
I hope this helps.