There are many ways you could word this. As for which option to choose, try omitting "and his wife's" and see what happens:
Read about he amazing story.
Read about him amazing story.
Clearly, both of those are wrong, and they can't be fixed by adding "and his wife's".
So, first let's fix the short version somehow. These are all viable candidates:
Read about him and his amazing story.
Read his amazing story.
Read this amazing story about him.
Now, let's put his wife back into the narrative:
Read about him, his wife and their amazing story.
Read his and his wife's amazing story.
Read this amazing story about him and his wife.
Of those last three, the middle one sounds a little off to me. I'd choose one of the others on this list, or reword it in some other way.