Since you only want to play melee netplay and produce music, I would recommend buying a used/refurbished desktop with a modern CPU that has high single core performance since that is what determines dolphin performance and it's also what's important for music production applications (according to some quick research I did). Depending on how important music production is for you and whether you are willing to buy used/refurbished, you could of course look at something new instead.
One thing that's very important for music/video production is an SSD. It's where you would store your operating system, programs and all the music/video files for your current projects as SSDs are much faster than HDDs and will improve the overall responsiveness of your system and reduce loading times for accessing music tracks.
Getting both those things does not cost a lot,
this refurbished PC looks pretty perfect for your purposes and it's less than 200$.
Now, that PC alone would be sufficient for playing netplay, even though it does not come with a graphics card - the CPU's integrated graphics are enough to play netplay just fine. Unfortunately, most integrated graphics chips cause small amounts of lag - starting from about half a frame to a frame on average, according to Kadano's tests. So you could consider buying a graphics card to minimize lag.
For your purposes, a GTX 950 or GTX 960 would probably be ideal as the GTX 900 series has been shown not to introduce any lag. The GTX 950 or 960 in particular are affordable and low power, meaning you do not need to upgrade your power supply to use them. Installing them in your system is as easy as undoing a screw and slotting them in place. You should be able to get the GTX 950 for less than 70-80$ used. GTX 960 will be slightly more expensive but is more capable at playing games. You will find the best deals on your local craigslist or similar used-goods-marketplaces/apps.
Lastly, regarding monitors - the BENQ RL2455HM was tested to have particularly low input lag and should be below or around 150$. In order to have even less input lag (a couple ms), you could also buy a CRT monitor for the best experience. These are usually 0-30$. Tell me if you want more info about CRT monitors.
