The only way to view private photos on Facebook is also the simplest... You need to get the user to accept your friend request.
There is a simple trick to get people to accept your friend requests though, I've tried it a bunch of times:
Most people are pleased to see that they have Facebook friend requests, and when they get a notification that someone wants to add them, they will check out who that person is:
About 90% of the time, the decision to accept a friend request is made from this screen, when all that is known about the user is the name and profile picture.
So, in theory, all you need to do is work out the name and image of a person that the private user would accept a friend request from... And pretend to be them using a fake profile!