The bad news is that, technically, you can't really view private Instagram users without following them.
The good news though, is that you can view private Instagram users without revealing your identity!
The trick is pretty simple, and is described in detail here: //www.blurtit.com/3942448/how-to-view-private-instagram-photos
Alternatively, if you're not interested in hiding your identity, you could use some simple psychology to get them to follow you instead!
How to get private Instagram users to follow you
I managed to get a major celebrity to follow me on Instagram simply by posting cool pics and mentioning either her, or hashtags she had recently used, in the comments.
It took quite a while, but eventually she took notice.
If the person you want to view private photos of isn't a celebrity, then hopefully this form of flattery will earn their trust a lot quicker.
Another cool trick is to target the people the account is following.
Engage with their friends, post some funny comments, get the private user to take notice of you by how much all his/her friend's are talking to you - the friend request nearly always comes through after a few days.