Hmm. Light on documentation but architecturally well written. The OAuthConnector module for Drupal has a long issue queue and few commits. But it works pretty well.

1. Install the list of dependencies (as long as my arm) drush dl connector oauth html_client oauthconnector drush en connector oauth oauth2 html_client html_client_oauth oauthconnector
2. Create a developer profile at Linkedin and retrieve the keys https://www.linkedin.com/secure/developer
3. Create a new Connector profile and add the keys in there admin/structure/oauthconnector/add
4. Add the Connector block to a region admin/structure/block/manage/connector/one_click_block/configure
5. Enable the permission for anonymous users to use the Connector admin/people/permissions#module-connector

Thanks for extra clues: sneyers-pelgroms.com A list of available fields. https://developer.linkedin.com/documents/profile-fields