Current Specs:
Windows 10 Pro 64 Bit
AMD FX-6300 6 Core Processor
NVIDIA GeForce GTX 1050 Ti
8 GB Ram
Definitely not the best specs, but at least I can run some games without too much lag. I can't really afford crazy expensive processors like the i7 so would an i5 be great or not?
An i5 is just fine to use and the core i5 9400f is a fast one. It should handle most games (besides maybe some edge cases) as well as the emulator can. This is assuming you have enough cooling and power delivery to keep the processor at a good boost clock. It will definitely be much better then your current CPU, but really any i5 or i7 3rd gen or higher from intel or any Ryzen CPU from AMD will give you a much better dolphin experience then your current CPU.
It should be good enough for Dolphin, unquestionably FAR better than what you have now. You might have to take advantage of the Turbo though, 2.9ghz is suitable for most of the GC/Wii library but a couple of the most demanding games might need a bit extra (the Rogue Squadron games for instance might have a few performance dips). I don't believe the 9400F can be overclocked, though there may be a way to force it to run all cores at the higher Turbo boost speeds. Don't quote me on that however, i'm not familiar with dealing with non-overclockable Intel CPUs.
There are a couple of other options if you're okay with AMD chips. The new Zen 2 based Ryzens are quite fast, as fast if not moreso than even the 9th gen Intels apparently (compared at similar clock speeds). One of the options is a bit more expensive than the i5 9400, the Ryzen 5 3600 which is $200. It's a 6-core, 12-thread CPU and can be overclocked (with the right mobo and cooler). Should be possible to get it to 4.2-4.3 ghz or so. This should net extra performance over the 9400, and would be useful if you plan on ever doing anything besides running Dolphin. I gather even Yuzu and RPCS3 run rather well on the new Ryzens.
The other option is not out quite yet, but AMD is supposedly releasing a Ryzen 3500 sometime this month. It's supposed to be targeting $150 or so, and I assume it will also support overclocking. You might be interested since it seems to be around your price range.
Intel does have the 9600KF as the next step above 9400, but it's even more expensive at $240. It has no iGPU, but a considerably higher base clock than the 9400. Since it's a K chip however, it can be significantly overclocked even higher with the right mobo and cooler. A lot of people have managed to get 8th and 9th gen Intel K series chips to at or around 5ghz. But again, it's pricier than the 9400 or even comparable Ryzens.
Well in that case im switching to your option with the AMD Ryzen 5 3600.
That is a good choice, the Ryzen 5 3600 is a bit overkill for dolphin, but if you want to get in to emulating even more demanding systems (CEMU, RCPS3, Ect.) then the extra performance and extra threads will serve you well.