Can I add a class that is already full?
Yes, but only with the Instructor's permission.
Whether the Instructor chooses to add you will depend on a variety of factors ranging from whether or not they have an actual open seat to whether the course seems appropriate for you or you have met the prerequisites. They do not have to add you simply because you ask.
If the Instructor agrees, you must obtain a pink add/drop slip from your advisor or departmental office, fill it out, and have the Instructor sign the form. If the end of the first week of classes has passed, you also need your academic advisor's signature.
Then, take the signed form to the Student Scheduling Office in 10 Enrollment Services Center.