The short answer is: There is no way to view private photos on Instagram.
There are a lot of apps, YouTube videos and other waste-of-time internet trolls that state the opposite: That you can in fact "hack" Instagram and view images or profiles that have been set to private, but the truth is you cannot view private profiles, and it is very unlikely that hacks will bypass the Instagram privacy settings for long enough to reach you.
Instagram (and parent company Facebook) have a really robust anti-hacking team, with some of the most talented security experts and developers in the world.
If you think you are going to outsmart them with some $0.99 app you downloaded from the black market app store, then think again!
There is no way to see private Instagram photos without explicit permission from the user in question.