What sort of games? It would really depend how the game(s) actually operates, and likely how you are botting.
If your bot sits on top of the game and does screen scrapes, and moves your mouse around, then you're looking at a very challenging task, especially if the game is full screen (requires minimizing to drop focus). If you can instead bot by intercepting packets, and sending back packets (no mouse action required) your requirements are much much lower.
If you are talking about something like runescape, where there's minimal hardware accelleration on low settings, and it's within a browser, you can probably run several on a server, assuming you have enough memory and cores to do it. (anything with software rendering is essentially gpu agnostic)
Renting first seems like a good idea, just to see if you can. Buying a server to see if it works is a lot of risk up front, if you rent a server for a month, and it doesn't work the way you want, you can stop renting the server.
Also, shame on you, botting is bad.