You don't have to get a new CPU, as as I said, it's game dependent. Some games will be bottlenecked by your GPU well before your CPU, but other games will have something more along the lines of Borderlands 2 CPU bottlenecking going on. I'd be surprised if something like Crysis 1 was even playable for you, as with my crappy FX-6300 it's nowhere near sufficient according to my standards (it drops below 30 fps quite often, but on average hovers around ~35 fps. For my personal preference, I consider this "unplayable"). But like I said, on GPU bound games at lower settings and a resolution less than 1920x1080 you should be fine.
However, in the event of a CPU bottleneck which you will probably stumble across when you're playing, you should know that lowering settings will likely not fix your problem. When you're playing a game and you experience a GPU bottleneck because the eye candy is too much for your graphics card to process, you can turn off this aforementioned eye-candy through the settings. For example, lowering the resolution from 1920x1080 to 1280x720 may mitigate or even eliminate a GPU bottleneck and give you acceptable frame rates again, so in that sense, GPU bottlenecks are much easier to deal with. CPU bottlenecks are not so easy, because most of the settings in a game reference abilities on the part of a GPU, so simply turning off or turning down these settings will only make it easier on your GPU, but your CPU bottleneck will still remain.
To give you an example, I'll go back to my issues with Crysis 1 and my bottlenecking FX-6300. I have a GTX 770 which is more than capable of handling Crysis 1 at 60 fps on very high settings for the most part. However, my FX-6300 rarely is able to keep up. In one area I was playing on in the game, my frame rate was stuck at 32 fps. My settings were at very high with 4x MSAA and at 1920x1080. To confirm if this was a CPU bottleneck, in the same area I lowered my resolution to 1280x720 (which is 44% the number of pixels on screen as 1920x1080), turned off all AA, and changed my settings from "Very High" to "High" and I still only averaged 32 fps. That's how hard it can be to deal with CPU bottlenecking, so what I'm saying is, even if you don't expect to game on max for certain games such as borderlands 2, even on medium or low settings you might get unacceptable frame rates.
As for JuX's statement about an FX-6300 being a mistake to pair with the HD 7770, that's not true at all. How would it be a mistake? With this CPU you would see far better frames in games than without. CPU bottlenecks would be a lot less frequent over your Athlon II X2 270, and ultimately it's still a very underpowered CPU. There's nothing anywhere that states or implies you should be pairing it with a high end graphics card. How would I know? Probably because I have a high end graphics card and an FX-6300, and I regret the decision to get one because at the time I had overestimated its potential. It's extremely underpowered, and it's without a doubt not meant to be paired with "a 256-bit 2gb ddr5 card with core clock on 1100mhz and memory on more than 6000mhz..". Ultimately, it'd complement your Radeon HD 7770 very well.
That being said, if you do upgrade, I don't recommend you get one. Now first, let me just say that there's not necessarily any reason to upgrade. If you see your frames are fine for most games, then stick with what you have. But if you see that your frame rate is a little too much on the low side, it'll very likely be due to a CPU bottleneck. If that becomes the case and you want to upgrade your CPU, you'll probably need a new motherboard. What's your current motherboard socket? AM3+ or AM3? AM3 motherboards supports some Athlon II series CPUs as well as the now discontinued Phenom II series. AM3+ motherboards supports the former 2 CPU lines in addition to the newer FX series. If you already have an AM3+ motherboard, then you should probably upgrade to an FX CPU. If you currently only have an AM3 motherboard, then you'll need to buy a new motherboard in addition to a new CPU.
If you do have to buy a new motherboard, I recommend you go Intel. I regret the decision to go AMD. Their CPUs are completely trumped by Intel CPUs. Combined with an Intel motherboard, if your budget is low I suggest you go for an i3-4130. It's about $100 - $120, the same price as the FX-6300, but in addition to outperforming the FX-6300 it's more future proof because much farther down the line you can upgrade to an i5.