Due to Instagram privacy, most of the profiles who don't want to let people view their private posts,photos etc hide them using Instagram private profile function.Once the profiles are private, there are only 2 option left that can let you sneak their private profiles.

                                                    

Yes, create a fake but interesting profile, adding some post,video or photos to the account and to improve effect of the profile, add some free follower and free likes using addmefast.com. Now follow the target profile and sit back untill the account owner add you. Well, good luck with that.

Yeah, some sites are providing such service where you need to input your username and respective website will show you private posts.  

Here is an example website - http://hackpremium.com/view-private-instagram-no-following/