How to show graduate schools my strong self study background?
Yes, things like typing up notes, writing a "learning/seminar blog" and being active on SE sites are good ways to help indicate the level and seriousness of your studies.  However, unless you are spectacular at this and manage to become well known this way, on most admissions committees, no one will have seen this stuff before they look at your application.  This means that they can check out your blog, notes etc but probably will not unless they are interested anyway, or it gets hyped up by your letter writers.  Consequently, this will be of some advantage in alleviating concerns of your time off, and is certainly worth doing for your own preparation, but it is better if you have something more reliable in your application.

Here is one suggestion: if there is an appropriate professor from your undergrad, preferably one you were fairly close with, you can try contacting them to let you know what you're studying, with aims to go to grad school, and ask if they can spare a little time to (i) suggest reading materials/exercises/projects, and possibly (ii) answer occasional questions you may have.  They may also be able to (iii) give feedback on your notes/blog, though it makes more sense to wait to ask about this until you have something in hand to show them.

Professors are busy people, so someone may or may not have/make time to help you, but we enjoy working with good students, so at least you should be able to get some suggestions on reading materials.  On the other hand, there's also the possibility the professor is willing to work rather closely with you.

In addition to providing valuable guidance to your own studies, this will keep at least one professor abreast of your continued studies, which should then come through in your recommendation letter from this person.