Your current CPU is not that much slower than the I5 8400 or the R5 2600, though for only $30 more I would suggest the 2600x as it comes with a much better cooler, has much higher boost clock of up to 4.25GHz on two cores and has XPM which if you have high quality cooling can boost the clock up to 200MHz more.
So it's much better in lighter threaded applications than the 2600, in fact at stock its over 20% faster. Obviously you can OC the 2600 up to 4.1GHz easily and use the $30 that you save to buy an aftermarket cooler, there are some quite good ones for $30 and $40.
Ultimately your I5 6400 is not bottlenecking your GPU that much, if at all. It would depend on the games, but I think the large majority of games would have little to none of a bottleneck.
If you are going to "upgrade" then you need to look at R7 2700x or I7 8700k or at least the 8600k. Going for a 8400 or a 2600 would be more of a side grade, sure you are going to gain 10% to 15% performance, but in terms of fps that amounts to like 3-6fps gains, hardly worth it investing $200 in a CPU and $100+ in a mobo.
The bottlenecking is real for all high cpu demanding games, the weakest aspect of the 6400 is not its c/t count, it is its lower turbo boost.
I see that clearly in AC Origins, as I said before, doesnt matter the resolution, even 720p at very low settings, I still get drops going under 45 frames.
Same as "low cpu demanding" 4k very low which the cpu still runs at 100% 4 cores and delivers the same results of 1k and 2k.
On the other hand, getting the 7700 or the 8400 (quite similar performance) paired with a 1070 + card wields easily 80+ stable frames.
Other comparisons of the 7400 with the 8400 shows, at 1080p, +20 - 25 frames with various games.
Anyways,
I got the 8400 and a B360 for R$ 1500.
My old kit should worth R$ 700-900.
In this upgrade I am going to invest about R$ 600 (U$ 150)