I have the best luck buying a standard OFFICE/BUSINESS PC (Made to withstand a lot of being on 24/7 etc) and then upgrading it!
RAM, vid card, and power supply. then you're pretty all set.
I did this same sort of thing.
To OP,
I bought an HP Pavilion (making sure the case was easily opened) for ~400 bucks, then just upgraded the power supply and GPU.
Look for one that has 8GB of RAM + a good CPU already, saves trouble.
I dont think a Office/Business PC are optimal choices. There is no point on leaving on a gaming Pc for 24/7. Sure, you can change Ram & PSU but they aren't build for gaming, so their CPU isn't optimal nor is the Mobo. GPU could be a 50/50 depending on if it fits with the case or whether the GPU is going bottle neck because of the CPU & etc.
It's a option but you'll have to probably downgrade on some parts because Premade Business/Office computers aren't going to have power.
Just my 2 cents