You should see some improvement, but I would imagine a C2D would pretty badly bottleneck a R7 270X.
Mouse's recommendation of a FM2+ motherboard (get the FM2+, not FM2) and an Athlon 750K/760K (I would go for a 760K personally if you do, as it has a few enhancements over the 750K) will give you much better performance. You could also go for an AM3+ socket motherboard and a CPU like a FX 4300 or 6300 (both of which have L3 cache which can help with better FPS in games), but the AM3+ socket appears to be dead, meaning options for future upgrades without having to replace the motherboard don't look good.
In summary, if you're going to buy that card, you should really upgrade at least your motherboard/CPU/RAM as well. A new PSU probably wouldn't hurt either. An Athlon X4 760K is a nice budget CPU, and paired with fast RAM (1866MHZ) will perform just fine for gaming (I know, I have one).
The AMD FX 6300 is the sweet spot for the AM3+ socket, but like I said, it's apparently pretty much dead at this point with respect to future upgrade options.
On a side note, you may want to hold off upgrading for a while, and wait until AMD bring out their FM2+ based Kaveri/Steamroller Athlons. They should be launching within the next few months, and offer a performance boost from the current 760K.