There are many ways to REGISTER the copyright of your weekly strip. Your strip is actually considered copyrighted from the moment you have created it. The registration process is different. It serves as evidence of your copyright and provides you with some legal protections that you might not be able to receive without registration, and may make actions against infringers somewhat easier.
REGISTRATION involves filling out one of the copyright application forms and sending it in to the Copyright Office of the Library of Congress. They will send you forms appropriate for your type of work. According to an inspector at the Copyright Office I spoke to recently, there is no one "correct" way to apply for registration. He said there are almost as many ways of doing it as there are people applying. As long as you submit the appropriate amount of examples for the type of application, the filled-out application, and your fee (which is $20 per APPLICATION, not per strip) you'll get your registration.
Your registration can be for as little as one strip per application or as many as a year's worth, depending on how much protection you want for all of your work and how much work you have available to register at any one time. It varies from cartoonist to cartoonist. We have been registering our strip at a rate of four weeks worth of dailies and Sundays per application, although we may change that to just under three months worth at a time. We'll be discussing the ramifications of that change with our attorney.
Which brings me to my last point. If you want to register, see an attorney first to get you started on the right foot and determine which way works best for the way you work. You can do all of the actual application work yourself. You don't need an attorney for that.
Best of luck,