Create an Instagram application here, and enter the data about your app (for redirect URI use something like http://your-redirect-uri?code=CODE). After your app is created, make sure that Disable implicit OAuth is checked and that you’ve added yourself as a Sandbox user.
Where YOUR_CLIENT and YOUR_REDIRECT are your CLIENT_ID and your REDIRECT_URI (without parameters)
Now you should see a login screen, login and approve (as the sandbox user). It should redirect to YOUR_REDIRECT_URI#access_token=SOMETOKEN
You now have your access token for viewing your own feed.
If you would like to show additional feeds of other accounts to which you do have access (e.g. your friends feeds), you can invite them as Sandbox users to your app (make sure they accept this invitation). You can then use this API token to show posts from their feed as well.