Making a link to create a login button is really simple.
All you need to do is make a link on your webpage/app
E.g.
‹a href=“https://api.twitch.tv/kraken/oauth2/authorize?response_type=code&client_id=[YOUR_PUBLIC_CLIENT_ID]&redirect_uri=[RETURN_URL]&scope=[SCOPES_REQUIRES]”›Connect With Twitch!‹/a›
Once the user clicks the link they goto Twitch. After they approve it redirect back the RETURN_URL you specified and you can process details such as auth key and run futher API calls using it as reference.