I think the very fact that you're writing stuff shows you're interested. I don't see the point in communicating with them every week unless you happened to have something worthwhile to say every week. Like you, I just sent in a LOI (was told by admissions to send this to the Dean) and then a final update letter (with multiple significant updates+final grades, told to write this to the committee in general).
I can think of some special situations when someone on a waitlist might want to talk to the Dean weekly; for instance if they're holding an acceptance at a school that starts very early and they want to know some inside info on the movement to plan out what to do (stay on the waitlist or withdraw). Or maybe they give update the comittee weekly with just one update instead of writing one big composite letter like I did.