For the purposes of you're question I'm going to assume a writing club is essentially the same as a writing group. If it's not, take whatever works and leave the rest.
#1 - Lay down ground rules. This is the most important thing. Come up with a brief document that outlines how people will critique one another, so everyone is on the same page. For instance, is this group mostly about support and encouragement? Or is it more about seriously improving the craft? Neither one is "right," it just depends on what you want from the group. You should also make some guidelines for respecting one another - that critiques need to be constructive rather than insulting or personal. One rule my group has that I really like is that nobody may assume that a particular work is a reflection of the writer's life or beliefs. (For example, if I write a story about a girl who has an abortion, nobody gets to imply or even ask if I've had an abortion.)
#2 - I don't think it's necessary to limit content to one genre or area, but it is helpful if everyone has sort of the same goals. What I mean is, you probably don't want one person submitting five pages of poetry and another submitting 50 pages of a memoir. I think for purposes of critique, the best thing to do is have a page requirement - ie, everyone can submit up to 30 pages per session. Or whatever.
#3 - If you're going to focus on critiquing, you might want to have people submit a piece of work before they can join the group, to make sure they are at whatever "level" you deem appropriate. It's kind of annoying to have everyone be good with writing mechanics and now focusing on the "finer" points needed for publication, and then one person who struggles with punctuation and grammar. Not that this can't work; again, it depends on what you want.