I only want to jump in to hardily recommend nvidia for linux. The proprietary driver installation is a pain, but once in place it is a great deal more stable, feature rich, and higher performing than AMD with OpenGL. I know you are putting your eggs in that vulkan basket so this advice might not be well received... after years of hacking away with AMD in linux myself, I have now sworn off the red team for good (at least for linux). Just wanted to share my experience.
As far as the rest of your build, if you are most interested in emulation, then I would recommend getting an i3 kaby lake processor and a good after market cooler. Even with a locked multiplier, and even on a cheap motherboard, you can achieve very respectable clock speeds by increasing the system bus speed. 4.5ghz stable should be quite achievable. Most emulation is single threaded or dual threaded, so with two physical cores and two virtual cores you won't be missing out on anything compared to the more expensive and power hungry i5s (I think PCSX2 officially supports up to four threads, but stability can be suspect and performance doesn't always scale).
As to i3 vs pentium, generally you see higher IPC with i3 processors because of the increased CPU cache and support for newer and more advanced instructions, in particular AVX and virtualization support. Seeing as how most emulators are quite CPU bound, I would recommend spending a bit more to get a bit more in this case. If you can afford it on top of that, a nice 3000mhz DDR 4 dual channel memory kit will cost you about 25% more than a cheap 2133mhz kit, but will help out a lot with some of the worst case scenarios where the emulator has to retrieve something directly cached in main memory.
Linux usually feels pretty snappy on a regular HDD if you have to choose between having storage only or a small SSD. A cheap 256gb SSD will only out you about 60 or 70 dollars USD, and then you can pick up a 1tb HDD for another 60-80 at some point down the line.
If you add in a cheap case, a half decent power supply, and 100-150$ for a decent GPU, I would say a realistic budget is about 650$. With those specs, 1080p in dolphin would be easy, as would achieving play ability in any of the more demanding emulators.
As far as the rest of your build, if you are most interested in emulation, then I would recommend getting an i3 kaby lake processor and a good after market cooler. Even with a locked multiplier, and even on a cheap motherboard, you can achieve very respectable clock speeds by increasing the system bus speed. 4.5ghz stable should be quite achievable. Most emulation is single threaded or dual threaded, so with two physical cores and two virtual cores you won't be missing out on anything compared to the more expensive and power hungry i5s (I think PCSX2 officially supports up to four threads, but stability can be suspect and performance doesn't always scale).
As to i3 vs pentium, generally you see higher IPC with i3 processors because of the increased CPU cache and support for newer and more advanced instructions, in particular AVX and virtualization support. Seeing as how most emulators are quite CPU bound, I would recommend spending a bit more to get a bit more in this case. If you can afford it on top of that, a nice 3000mhz DDR 4 dual channel memory kit will cost you about 25% more than a cheap 2133mhz kit, but will help out a lot with some of the worst case scenarios where the emulator has to retrieve something directly cached in main memory.
Linux usually feels pretty snappy on a regular HDD if you have to choose between having storage only or a small SSD. A cheap 256gb SSD will only out you about 60 or 70 dollars USD, and then you can pick up a 1tb HDD for another 60-80 at some point down the line.
If you add in a cheap case, a half decent power supply, and 100-150$ for a decent GPU, I would say a realistic budget is about 650$. With those specs, 1080p in dolphin would be easy, as would achieving play ability in any of the more demanding emulators.