More years ago than I like to remember I was involved in running Heritage Society, the Oxford University Folk Society. But since we were the only folk club in town, we didn't worry too much as I remember whether people involved were students or not.
It all seemed simple enough - a stall at the Freshers Fair, a entry in the relevant lists of societies, a small committee to take care of business, such as making arrangements with a pub for a room for a weekly session, and booking guests, and MCing the sessions. In between the sessions, people would hang out together and play music and arrange trips and so forth.
You've got a couple of existing clubs in Cambridge, so I take it there's something lacking in what they are doing. So I suggest you just go ahead and do whatever you think needs doing to fill in that gap.
Have fun. (And I'm only down the road a few miles if you ever feel like inviting me along.)