Not sure if it's legal so I can't post it here but there's one website I know that creates hacks like that. They're always the first in finding website loopholes, especially instagram related. If you want to know the name of the site, leave a comment here and I'll message you.
The best way to view private Instagram photos is not to install an app, but to perform a very simple trick:
Do some research on the user you want to connect with. Check their other social profiles such as Facebook or Twitter, and find someone they communicate with often.
Make sure that person is NOT on Instagram.
Then create a fake Instagram profile as that person, using photos of that person taken from other social accounts.
Once your account is looking genuine, friend request the private account as if you were this other friend of there's just joining Instagram. And voila! You should have access to the private photos on their Instagram.
If you can't find a friend that matches the criteria, just pretend to be a fan of someone they really admire. For example, if they love 1D, then pretend to be a super 1D fan too and post loads of their photos on Instagram.
It might take quite some time, but it's a genuine way to view private Instagram photos.
For some reason, most hackers of Instagram private profiles seem to explain how they access people's private data.
Even though that's kinda risky (because I'm pretty sure that kinda thing is illegal), it's also great because it means you can work out how to do it too.
How to view private instagram photos.
Below I've listed some of my favorite YouTube tutorials, all made by friends of mine actually, showing you how to make use of glitches, software and site errors to access a user profile on Instagram.
They even work when they haven't given you permission to private Instagram photos (and let me tell you, that's where you'll find the juiciest images).
One girl from my old school in Minnesota got hacked and sexting pictures she was sending to a teacher got around the school!!!! I was like WTF OMG :OHow To See Private Instagram Profiles 
How To Hack Instagram
Instagram Private Profile Viewer (Updated) (Working January 2014)