Private users on Instagram don't really make sense to me. Instagram is all about 'sharing' and 'being social' - you may as well keep your photos on your phone if you don't want anyone to see them.
Anyway, if you're hoping to access someone's private Instagram account without already being friends with them, you'll probably want to read up on the latest news on Instagram hacks:
How people gain access to private Instagram profiles
A hacker called phr0nak posted an article on Reddit about how he managed t find a security flaw in Instagram, and was able to see private accounts: http://www.reddit.com/r/netsec/comments/1xiiin/how_i_hacked_instagram_to_see_your_private_photos/
He goes on to say that he alerted Instagram to this flaw, and received a reward for his honesty and good work.
Since then, Instagram have really beefed up their security on the platform, and it's looking pretty impossible to get access to someone's account without consent.
Why is it such a big deal?
Because actually, if Instagram doesn't aggressively defend its users from hackers, it could land itself in legal trouble.
Here's a headline about similar problems that messaging app Snapchat is having (via Zdnet.com): http://www.reddit.com/r/netsec/comments/1xiiin/how_i_hacked_instagram_to_see_your_private_photos/
