You unlock the ability to send in Strike Teams, after gaining access to the Tempest and speaking with Kandros on the Nexus.
You start out with 1 Strike Team to send on missions with the ability to recruit more. The first time you recruit it's free, then your second team costs 40 mission points, with each proceeding team increasing by 40 points. (0, 40, 80, 120, 160, 200)
When you choose your missions, some of them will give you the ability to send a Strike Team or play in multiplayer. Do whichever you'd like. Winning or losing will count towards your total number played. Completing 5 will net you this trophy.
You can also send your strike teams using the Apex HQ app which can be downloaded for iOS or Android. I can confirm that you can send out your strike teams via the app and it will count. You will not gain the trophy from the app, you will first have to go online via the game and log in from the Tempest, at which point the trophy should unlock for you.