1. Go to the classes anyway, never missing one. Ask the prof each day if anyone has dropped. Once a student has dropped, get his/her permission to add the class (the form will usually require a signature from the instructor if it's past the add/drop deadline)*.
2. This isn't the most ethical thing to do, but it's often done: If you have a friend at the university who can enroll now, get him/her to sign up for the class now, "reserving" it for you until you can enroll in July. Then he/she can drop and you can add it right after, before anyone else snatches it up.
*My Better Half (MBH) is teaching calculus at a local university this quarter; because of budget cuts, many class sections have been cut, leaving intense competition for the remaining sections. The seats fill up very, very quickly, and not everyone who needs the class can get in. However, one student faithfully attended each and every lecture this quarter, and even took the midterm; fortunately, after the midterm, a few students dropped the class and she was finally able to officially enroll in it.