Demonstrating iOS in-development application
The most common way of doing this is through TestFlight, and you will need an Apple Developer Account.  

Setup your customer as either an external or internal user.  Upload the app to TestFlight.  The customer will be able to download the app and test it but it will not be available for the general public.

Of course, there's a bit more to it than what I just listed but it's not very difficult and you can upload the app multiple times (after each bug fix or app change) and the customer will be automatically notified to download the new version.  When you're ready to push it out to the public, you can do that.

Go here for more info on how to use it.