The client certificate is used to identify the client. If the programmer of the app made his job well, you will face difficulties (hopefully). Likely, and most secure, the private key and identity resides in the key-chain. Less secure, it resides in a secured archive (.p12, .pkcs12, .pfx) in the bundle, whose password resides in the key-chain.
If the programmer did his job not so well, you might find the password of the secured archive in the clear somewhere in the apps binary (there're actually floating samples around which do exactly this).