The following link, Techpowerup.com has benchmarks for AC4 for various GPUs including the R7 260x.
https://www.techpowerup.com/reviews/NVIDIA/GeForce_GTX_750_Ti/7.html
The test rig uses an i7-4770k for all benchmarks. According to the benchmarks, at 1080p and max graphics the R7 260x only manages to get 20 FPS. Based on the benchmarks from my previous post. A GTX 770 paired with an i7-4770k managed to get an average 49 FPS while the FX-4300 only managed to get an average of 35 FPS with the same card. That basically means, the FX-4300 only provides 71.4% the performance of the i7-4770k based on these benchmarks. This basically means that a FX-4300 with a R7 260x would get an estimated average 14 FPS at 1080p and max graphics in AC4.
Looking at the benchmark numbers, dropping the resolution from 1080p down to 1600x900 should give about a 25% increase in performance for the R7 260x. That basically means with an FX-4300 the R7 260x's performance should increase from around 14 FPS to around 17 FPS. If dropping the graphic settings down does not improve the performance by much, then you need to further lower the resolution to 1366x768 to get better performance.
If 1366x768 and low graphic resolution does not improve performance to your satisfaction, then you need to either upgrade the PC or buy a new PC with much better performance.
Generally speaking, Intel Core series CPUs performs much better at games than AMD FX CPUs. In the benchmark chart from my 1st post, a dual core i3-4340 CPU can outperform AMD's 8 core FX-8350 CPU. Although a quad core Intel i5 CPU is generally recommended since some games do make use of more than just two cores. A Core i7 CPU is not really necessary to play games with. It is similar to a Core i5 CPU, but it has something called "Hyper Threading" that games generally do not benefit from unless the CPU only has 2 cores, like the Core i3 series.
As for a potential graphics card, that depends on your budget. However, you can click the following link for some benchmarks of modern GPUs that are not too expensive. AC 4 is not used as a benchmark, instead the newer AC: Syndicate takes it place.
https://www.techpowerup.com/reviews/Gigabyte/GTX_1050_Ti_G1_Gaming/
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