The rule of thumb for building a gaming computer is to pay double for your GPU than your CPU. (E.g. Spend $60 on a CPU, $120 on a GPU)
The FX-6300 should suffice when playing a majority of games, especially at this current date. Over-clocking it would get you better results.
The R9 270x is a good mid-range graphics card and will perform well when paired up with an FX-6300, so I'd suggest doing this.
HOWEVER, make sure that your PSU will support your new CPU/GPU. What PSU do you currently have?