Audio work has extremely low requirements. Every week I do audio editing at my church on a 5 year old i3 processor with no problems whatsoever. A faster HDD than the craptastic laptop HDD in the system would help export audio faster, but it really requires very little CPU and RAM to do audio work. Any gaming PC should be able to do audio editing without issue.
HOWEVER: Cheap gaming rigs tend to be noisy as hell. If you want to do audio work, then do youself a favor and get a tower with some acustic padding and quiet fans inside. Get a GPU with a nice quiet cooler, and an aftermarket CPU cooler with nice quiet fans, or look into water-cooling the both of them.
For gaming an i5 processor is all that you need, and that will be entirely overkill for any audio work you need to do. AMD has some decent options, but (at least for now) Intel is the way to go for gaming.
SSDs are nice, but even a slow HDD these days is more than fast enough to record several uncompressed audio streams at once in real time. Absolutely get a decent sized SSD to hold your OS, programs, games, and scratch disc space... but for archiving and recording a nice fat HDD will save you money and offer plenty of space while still offering more than enough performance.
Your GPU has nothing to do with audio. I have always used nVidia GPUs in all of my recording rigs and never had a problem. Again, whatever GPU you decide on, be sure to pick one up with a nice quiet cooler, but choose your brand based upon the games you play, not your audio workflow.