Well it looks like it's possible to build something like I wanted for 500 to 600 bucks. It would be a lower end gaming computer by today's standards but way more than what I have now. Plus there's the fun of putting one together. Probably end up getting something like marriedman said.
https://m.newegg.com Love this site. Used it last time I built gaming computer, but it was years ago...like 2008. Virtually build your computer. Lays it out for you. Compare shizzz. Surely it hasn't changed much. Plus if you need tires, just put those in your cart.
Looks like you got a lot of good suggestions, the mathematical combinations are astronomical, have fun choosing! Games are far more demanding of a PC than any office application, email or web browsing, so get a gaming magazine and see what the crown is using. I was never into gaming after working on them all day long! A lot of satisfaction in doing it yourself.