Follow Client (Implicit) Authentication . Use the credentials from an application registered on the Instagram developer portal. Therefore, change the CLIENT-ID and REDIRECT-URI from this URL to the credentials of the registered application.
From your application, send the user to this link. If they agree, they will be redirected to uri redirection (which should be somewhere in your application). They are redirected to the page, the page will contain a hash containing the acces token.
Get this from the url and use it to request data from instagram api. You will need jsonp for requests. See the example in action in a similar thread
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