The 5800K shouldn't bottleneck that GPU, some games just won't perform as well as they would with a faster CPU like say, a Haswell I5 due to the general efficiency and the ability of that CPU to extract more out of the system around it. If I remember correctly (Correct me if I'm wrong) the 5800K is a Piledriver based CPU (Basically an FX 4300) and performs slightly better than a Phenom II X4 965. I happen to have one of those old hogs at stock clocks in an older rig paired with an overclocked HD 7850 running at 1050MHz, and experience no real CPU bottle necking on that rig. The GPU always maxes out before the CPU does. If your CPU maxes first, you have the definition of a true CPU bottleneck. If anything, my GPU is the bottleneck in my rig. The R9 270X is basically a 7870, a step up from my example GPU, and is noticeably faster, but shouldn't be fast enough to totally overrun your CPU. I have no doubt my old Phenom could handle an R9 270X and would make gains in games if I replaced the 7850 with one.
As I7Baby said, if there is a bottleneck it would probably be marginal and not really noticeable. And there really is unfortunately no definitive answer as to what bottlenecks what until you test it. And of course, CPU bound games will benefit more from faster CPU's. But that doesn't mean your CPU is too slow for the GPU, it just means it's not as fast as a faster CPU. An I5 3570K would appear slower if compared to an I7 4790K in some CPU bound games, but that doesn't mean the I5 is a bottleneck. The 270X will be a large step up from your current GPU for sure, and would definitely give you a noticeable boost in gaming performance over the 7770. I'd say go for it.