This iOS 5 sample app clones the basic functionality of Instagram (sans camera filters and geo tagging). It uses storyboards and ARC (automatic reference counting).
First thing the app does on launch is present you with a login screen. It needs to authenticate you with Facebook. Once authenticated the app sends data to the Web App ( see below ) and the web app creates you and/or logs you in.
git clone git://github.com/facebook/facebook-ios-sdk.git
cd % ~/facebook-ios-sdk/scripts/build_facebook_ios_sdk_static_lib.sh
This will create the static library under the <PROJECT_HOME>/lib/facebook-ios-sdk folder (e.g. ~/facebook-ios-sdk/lib/facebook-ios-sdk). You may then drag the facebook-ios-sdk folder into the app Xcode project to include the iOS Facebook SDK static library.
More info on this here https://developers.facebook.com/docs/mobile/ios/build/
Git clone the accompanying django web app I wrote if you want to actually login and upload photos. Read more about the web app here: https://github.com/ethanlance/instaclone_django/blob/master/README.md
git clone git@github.com:ethanlance/instaclone_django.git