They cost anywhere from $5 to $50 or more, depending on the provider and plan. I wouldn't recommend paying more for a big name like PhotonVPS for botting when there are plenty of cheaper alternatives. Photon might not go down at all in a month, compared to once or twice for a few hours for other providers, but that's the main difference, and for botting, not that big of a deal. I like Pritell (you can run Windows if you want, and the support was good), but there are several others with similar pricing. If you can, find a few reviews (Webhostingtalk.com or another site). Just make sure that the resources are dedicated, not shared (Don't get OpenVZ).
You generally access a VPS via a VNC Viewer, and there is a very good setup guide on powerbot (I believe it is also posted on RSBuddy, but I'm not sure).