Hello,
In response to your question, there is no current known way to view a private Instagram page unless you are following the user yourself.
Instagram has a very strict privacy policy that is invested in protecting the user from any violation of said privacy. Any loopholes or 'hacks' on this policy would result in a decline of the company's reputation and thus an inevitable collapse of the entire site. In knowing this, it is in the best interest of all users to respect the privacy rights of each individual in order to keep a functioning system and also to protect ourselves.
With that being said, the solution could be as simple as just requesting the user to view their page, and they may surprise you by accepting! Some people choose to create 'alternative' accounts to add private users with anonymously, however there is always a danger of being caught, which may not be worth the effort.
Good luck.
