You wont get a better GPU for the price of 270x.
Now an intel i3 + h81 motherboard will be mildly better then an fx-6300 and a 970 chipset board.
Difference being is the next step up to an i5 is much better then the 8350.
If you still want to stay AMD I would highly suggest getting an 8300 or 8310 from tiger direct for $20-30 more, it will be a very noticeable improvement over the 6300 for the very small increase in cost.