A club implies that there is some sort of official process involved before one can become a member.
"I want join your club" implies that you are not a member and would like to become one.
"I want to participate in your club" implies that you are a member but have been inactive, and would like to get more involved.
A circle is more of an "unofficial" club, that is, it is a groups of people with a similar interest or who live in a specific area that befriend one another and get together socially. There are no official club rules or dues.