$300 is a seriously low budget for a good pc. You might want to consider saving up at least 500 first. Anyways, you should go with the Pentium G3258 processor as it is Haswell (4th generation) and overclockable with a Z97 motherboard. It has serious single thread performance (which is what Dolphin requires, it relies heavily on the CPU, not GPU like most PC games). You can overclock it very high with the stock cooler and the air because it doesn't use much power and it will perform similar to the highest end processors in Dolphin. 4GB of RAM is sufficient, and as for the graphics card you can get away with very little since Dolphin doesn't need a good video card unless you want to play 1080p+, but in order to future proof your machine you should get something that isn't total crap, I would go with the GTX 750 ti. It's basically the best price/performance budget graphics card.
Basically, build your own computer. Prebuilt computers offer you crap hardware for overpriced prices and you can't tailor the computer to your needs. Get started on pcpartpicker or the buildapc subreddit. It's pretty easy to build it, the hard part is getting the optimal parts. Again my recommendation is a Pentium G3258, Z97 motherboard, 4GB RAM, and a GTX 750 ti. You probably wont get that for $300, but at least stick the with processor/motherboard choice, it's the cheapest and best for Dolphin.