Taking AP classes would show up in your transcript and show a slight rigor and an increase in challenge, though this wouldn't help much if the student got a low grade.
As for reporting self-studying, there's an "additional info" section on the common app. You can mention that you're self studying AP this and that.
It'd also be useful to take AP classes that you know will help you in college. Perhaps it'll let you get rid of a liberal arts requirement or let you skip an intro class and take an advanced one right away