The badge is actually awarded for working with one of the other contacts (Dubrowski, Ansaldo or Maros) during the story arc from Vincent Ross, and can be done via Flashback.
Read the mission text carefully, because it will tell you when to talk to your other contact. Each contact will give you an additional objective to complete in each mission on top of what Ross wants you to do. You must complete each of these additional objectives in order to receive the Connected badge.
When in doubt, just talk to the contact you chose before each and every mission to make sure you don't miss something, 'cause if you forget to talk to them or forget to accomplish their task in a mission, you won't get the badge.