Publishing code in order to improve my chances to be admitted to a CS graduate school
It sounds like you have a portfolio that should be a part of your application to university or a job or anything. Furthermore you're saying that the working product is a better testament to your skill than the code itself. This is also fine.

The only concern I see is how much work you will require on the reader of your application. If you link to a YouTube video, also include screenshots. If you link to a GitHub profile, consider writing up some basic statistics on the activity or health of your activity there.

Regarding where to publish code, don't overthink this. Github is popular, but a Dropbox file, a Google Doc, a file sharing service, or anywhere else you can post a file will do.

I do not know the proper channel to submit a portfolio or the proper format. This situation is probably documented in the application instructions however, so you should be able to find info there, or reach out to a recruitment officer for how to proceed.

EDIT regarding where to post - actually a .zip file or similar may be problematic since they tend to look like viruses to people, who don't want to open them. Github or a competitor may be the best choice since the source code is browsable in their web UI. It is best practice to include a software license whenever you publish (literally "make public") code, even if you think it is unusable. I think the MIT and Apache licenses both amount to (IANAL) "do whatever you want with it, just don't sue me."