Unless you're some kind of secret hacker that has skills way beyond the average internet user, it's not possible for you to view private Instagram profiles without following. As you've asked this question, I'll go ahead and assume you're not?
Why would someone want to view a private Instagram profile?
There are several reasons why someone would want to view a private Instagram profile:
- They don't want the person to know they're looking at their images, so won't send a friend request.
- They've been blocked by a user previously.
- The user doesn't want them to see the images so won't accept any requests.
My advice would be to send a request to the user to view their images. If they say no, just leave it. It's not like you're missing out on much anyway. From my experience, most people post pictures of their food. Who cares about that?!