The best way to access private Instagram profiles and pictures is also the simplest... Simply ask for permission from the account holder by friending them.
If they don't respond to your initial request, you can also mention them in comments making your request.
Just make sure you comment on a photo that they are likely to find cool or interesting, and make sure your profile matches their interests too.
What usually happens when you try to friend someone on Instagram is that they will check out your profile and previous posts to "vet" you, and make sure you are someone they would be interesting in connecting with via Instagram.
Use your knowledge of the person in question to make your profile and pictures as appealing as possible, and they should soon trust you enough to invite you to their inner circle of private Instagram content.