Greetings there.
Mike from Toronto, Canada here.
I've read your specification outline and I have extensive experience working with Instagram API.
Based on your requirements, here is what I suggest.
1. Build a website to allow you to enter a username into a search in order to gather information about the Instagram page.
2. By integrating Instagram API - we will be able to retrieve the information in real time and report it back into your website.
3. You will be able to set filter criteria;
a. Most Liked
b. Most Commented
c. Display in sequence on a per user basis.
By clicking on an image, it'll bring the user to the actual instagram post.
Private accounts cannot be extracted and be included in the webapp as it's restricted.
Send me a message on Freelancer to discuss further and view some sites of mine.
Take a look at my portfolio -340 + posted reviews.
Best wishes,
Mike