Email the professor and ask if you can be added to the class. Find the professor's email on the college's web site, or call information at the university if you can't locate any contact information. Let the professor know your situation, that you really want to take the class and are interested in the subject matter. Be professional in your email, without resorting to desperation tactics. Above all, make sure you proofread and spell-check your email, so the professor sees that you are a serious student.
Show up early to class the first day and try to talk to the professor before the class starts, to ask if there are any available seats. Frequently professors can create a "wait list" to which you can be added. When students drop a class in the first week -- and every class will have students dropping -- you will be on the list to be added in first.
If you're not able to talk to the professor before class, make sure to stand out in the class in a positive way. Sit at the front of the class and demonstrate your interest in the course by asking pertinent questions. Then try to be the first to talk to the teacher after class, before the rest of the students trying to add get there. Wait lists are usually populated in a first-come-first-serve manner.