Upgrade help needed

Bf1 is a pretty cpu intensive game. The best way to test what you've currently got is to use a program like msi afterburner to monitor the cpu and gpu usage while playing the game. If the cpu is 90-100% consistently and the gpu is only around 60-70% usage, the cpu is holding things back. The cheapest upgrade for you might be the fx 8320 or fx 8350. It could be pushing things a bit though with only a 350w psu.

Going intel would mean a new cpu, motherboard and ram if going skylake or a new cpu and motherboard while keeping the ddr3 ram if going with 4th gen like the i5 4590.

Bf1 can make use of additional cores/threads but it also relies on cpu speed and core performance. It's difficult comparing amd and intel directly, amd usually has higher clock speeds but intel processes more per clock so on bf1 an i3 which is only a dual core cpu with hyper threading performs similar to the fx 8370 (a bit better than the fx actually) despite the fx having 8 cores. Normally I wouldn't recommend upgrading with another fx cpu but it could keep costs down for you and still be an improvement over the fx 4xxx cpu you're currently using. So long as the gpu keeps up the fx 8xxx should be able to stay roughly around 60fps.

http://www.gamersnexus.net/game-bench/2673-battlefield-1-cpu-benchmark-dx11-vs-dx12-i5-i7-fx/page-2

A 3rd gen i3 is more comparable to the fx 4300 and often out performs it in games, not an i5. Price comparisons often reflect comparative performance regardless of cores and clock speeds. The weaker per core performance of the fx cpu's would normally make the 8 core fx not much of an upgrade but it may help a bit in bf1. If possible I'd suggest upgrading the power supply to at least a decent 450w unit either way. Even then a psu and fx 8320 or 8350 would still be less than switching to intel.

Just trying to give options without knowing your budget. Obviously spending a bit more and changing out more parts with a switch to intel will give more performance but whether it's worth it to you or not or doable on your budget is another matter. Also keep in mind that if switching out the motherboard you'll have to reinstall windows, if you're using an oem copy that isn't transferable like a retail windows copy that could pose a problem. While upgrading to an 8 core fx may not be as huge of an upgrade it can be done without reinstalling windows which could be seen as a perk.

Depending on what case you have and how much room it has to install an aftermarket cooler, an inexpensive cooler like the 212 evo or cryorig h7 would keep your upgrades to around $200. This may give you an idea.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-8320 3.5GHz 8-Core Processor ($129.88 @ OutletPC)
Power Supply: Antec 450W ATX Power Supply ($34.99 @ SuperBiiz)
Total: $164.87
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-12-11 06:02 EST-0500

With a decent enough cooler for overclocking a bit.
PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-8320 3.5GHz 8-Core Processor ($129.88 @ OutletPC)
CPU Cooler: CRYORIG H7 49.0 CFM CPU Cooler ($34.88 @ OutletPC)
Power Supply: Antec 450W ATX Power Supply ($34.99 @ SuperBiiz)
Total: $199.75
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-12-11 06:04 EST-0500