What does "dog's circle" mean?
I'm sure someone can give a better answer than mine, since I have not read or watched the series.

Dog's circle is not a Japanese phrase. Looking at the Japanese wiki article, it mentions an Inu no Kai (犬の会), or Gathering of Dogs.  It says it is a drinking club of youkai that would rush to Natsume like a dog for whatever reason he calls them, therefore the "dog" name.

I am assuming the "gathering" part of the title was translated as "circle", which I don't think is a very good translation.  Circle is used in Japanese to refer to clubs, like an "anime club", but does not translate over well back to English in that same sense.  I think the name should've been capitalized to make it clear it's a proper noun. Still, if they're throwing a term like that around, I would assume it was explained somewhere in the series.