i5 4xxx is still ok. Yes, there is some bottlenecking possible on processor intensive games like BF1 but it is not yet something I would be worried about.
Ón the other hand, since you have 144hz monitor, it could prevent you from having far over 60 fps on some games.
Updating the proc is up fully to you. I would say, not yet a must.
R9 390...hmmm... R9 390 is about on par with GTX970 and only slightly less powerful than GTX1060 6GB. If you play on 1080p, it is not yet a necessity to replace it. If you want to replace it, you should be looking at GTX1070 or Vega 56 as minimum. R9 390 is still ok.
There are some GPU intensive games to reach far above 60 fps @1080p, which is bottlenecked with this GPU but not by much e.g. Witcher 3.
Here, I also would say, you do not have to replace it yet.
Your PSU CX is not exactly the best but it is also actually not that bad. On the other hand, the CX is not really reliable and I saw several of those died and took the system with it. Changing the PSU would not increase any performance but...I will leave this fully up to you...
16GB is a wise choice. You have SSDs. Everything is fine here.
Basically, there is still no meaningful performance problem or bottleneck in your system. I would simply keep it as it is and go all out for a new system about 1-2 years in the future.
If you really want to upgrade your system step by step, I would start with the graphic card.
Note:
Games are differently written. Some are cpu intensive (A), some are gpu intensive (B).
Your I5 is a bottleneck for new heavy type A AAA games.
Your R9 390 is a bottleneck for new heavy type B AAA games.
all in all...you can still reduce the in-game settings to gain more fps