You can build yourself a decent pc for around that, better than you can buy prebuilt probably.
If you choose to go AMD id suggest a 4.0GHz 8-Core FX-8350 cpu for around $150 USD (i run 4 bots, 1 osbuddy account, youtube and other things in the background and has around a 90% CPU usage)
If you choose to go Intel id suggest a 3.7GHz i5 6500 cpu for around $200 USD
Since you aren't gaming GHz and core count is pretty important for multitasking like you want to do.
Motherboard you will need to choose according to your CPU since they use different chipsets, but basic boards are anywhere from $40-$90 USD (make sure you get a reputable brand though like MSI, Giggabite, Asus, ASRock etc)
Graphics card can be important as well since the more processes you have open the beefier GPU you'll need, even if you run your clients on Low CPU mode which reduced the FPS to around 10.
Id suggest like a RX-460 (which is what I use) and can be bought for $100-$130 USD (Sapphire is a pretty reputable brand and is on the cheaper side aswell)
Ram, I personally have 16GB but that's over kill so 8GB should do fine, costing around $50 USD
Cases you can get for like $30 for a cheap one, it doesnt really matter as long as it will fit the size of motherboard you want.
Harddrive, 1TB harddrive is around $50 USD
So it'll cost you around $500 on the high side for the above.