I don't necessarily see the point in calling an change from an i5 2320 to a 4th Gen i3 as an 'upgrade'. Haswell has higher IPC, better architecture, and probably a little more instruction sets but we're talking about 4 real cores vs 2 real cores and 2 HT Threads.
Additionally the i5 2320 is a 3GHz processor. It boosts up to 3.1GHz on all 4 cores, 3.2GHz on 2-3 cores and 3.3GHz on 1 core. But Sandy Bridge and Ivy Bridge processors can be overclocked 4 bins/400MHz from the Turbo Frequency. So in this case if the OPs motherboard allows this, the speeds would be 3.5GHz with 4 Cores, 3.6GHz with 2-3 Cores, and 3.7GHz with 1 Core. I think I've heard that some motherboards allow CPUs to run all 4 Cores at the Max Turbo speeds too.
Like many people said the R7 250 is an entry level video card. It wouldn't be bottlenecked that much even with an AMD processor. The i5 will be fine with even high-end GPUs. In a lot of games a 2.5GHz i7 will outperform or similarly perform a 8-Core AMD that is OCed when paired with a top level video card. I'm thinking it would be similar with an i5 also.