When you say "display a repo and its contents" you actually say "display the state of the repo after the latest commit of the master branch", right? That's actually the better way of thinking about the problem and will be a better guide through using GitHub's API.
You need to look at the Git data part of the API. Here's what you need to do:
1) fetch the list of refs for your repo using:
https://api.github.com/repos/:user/:repo/git/refs
Working example:
https://api.github.com/repos/izuzak/noam/git/refs
Notice that it lists the references in your repo and gives you links to continue.
2) fetch the commit object of the ref that interests you, namely "master", using the link provided in the response to 1):
https://api.github.com/repos/:user/:repo/git/commits/:sha
Working example:
https://api.github.com/repos/izuzak/noam/git/commits/5cf12775b844664d5f7af6663706195680181374
Notice that you get back an object with a link to a tree.
3) fetch the tree object of the last commit in the master ref, using the link provided in the response to 2) :
https://api.github.com/repos/:user/:repo/git/trees/:sha
Working example:
https://api.github.com/repos/izuzak/noam/git/trees/8a721bea8d2f281c87b39c74cbf5a70075d686b4
Notice that you get back a list of files in the root directory that is your repo. This is what you want. If you have subdirectories, you will get links to fetch the files in those subdirectories.
This should be enough to get you started :). Good luck!