My system isn't the highest end but can run stuff nicely. Most games 1080p60 at good details, including Dolphin running Project M at 1080p60 with netplay. I got a GTX 1050 Ti for an insane deal of Amazon during Black Friday ($130... seriously). I'm not sure if you will need the extra 2GB of VRAM the Ti offers, so I would suggest the 1050 unless you plan on playing actual PC games. As for PS2 games, a system that can run PM at 1080p60 can definitely crush a PS2 game.
As for the CPU, you don't need the best one. I have an i5-2500K running at stock speeds with the stock cooler salvaged from another computer. Something similar in performance would be good enough for PC gaming and Dolphin. You can probably get a somewhat cheaper CPU also (maybe a newer i3).
RAM isn't that much. 8GB works great for me, and DDR3-1333, however old it may be, works just fine for all my needs, Dolphin included.
Everything else doesn't matter much. Get some good drives, maybe an SSD/HDD combo or just a hybrid drive, or only an HDD if your budget is tight. Motherboard? Get one from a reputable brand that is compatible with your other stuff. Case? You can get the cheapest one what works, or get a flashy one with a window. Wii stuff isn't that demanding, so you probably won't be outputting too much heat, so the best airflow isn't that much of a concern, but a better case is great if you plan to upgrade or do other PC gaming. The PSU also - get one from a reputable brand with enough wattage for you and the connectors you want. DO NOT CHEAP OUT ON A POWER SUPPLY. A friend of mine wrecked his whole rig with a shoddy power supply. Would you rather pay $20 less now for the off-brand, or $500 later when you realize you have to replace your whole PC? Corsair, Thermaltake, and EVGA are among the good brands for a PSU. Make sure it has enough wattage for your system.
For picking parts, go to pcpartpicker.com. They automatically see if everything is compatible for you and gives easy buy links.