AMD Phenom II 955 @ 3.2ghz
Running at 1920 x 1080
It's your CPU that's the bottleneck. FF XIV is unusually CPU intensive compared to most modern PC games. You basically got a great video card in a system that's keeping it from it's full potential.
I'm running on a Radeon HD5850 which is a little less then your HD5870 in terms of GPU power. My CPU is a Core I5 750 (Quad) @ 2.66 GHz . I'm running both GPU and CPU at stock speeds - no overclock. In most open areas of FF XIV I'm seeing a silky smooth 60fps with FRAPS.
I built my system from scratch about 6 months ago for well under a thousand - a nice full-size HAF case, power supply, CPU, memory, video card, hard drives, etc. for well under $1000. You already have a case, hard drives, video card, and a power supply that's obviously good enough to power your HD5870. You can upgrade your motherboard & CPU to a nice core I5 or core I7 for a good price online - don't get the absolute "best" or "fastest" CPU, they are always horribly overpriced compared the next CPU in line. Just look for the CPU that's in the "sweet spot" in terms of price/performance currently. I think you'll need some DDR3 memory as well, but you can pick up 4 gig for pretty cheap these days as well.
That would be my recommendation for upgrade.