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. Completing 5 will net you:
I believe, given the wording on this trophy, that by 'Successfully Complete', it means that you have to win the mission and not fail it, 25 times. I will try to confirm that however unless someone else can.
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.
Edit: In the Multiplayer menu under Prestige -> Challenges -> Accomplishments, you can check your progress towards this under the Veteran Accomplishment. You will see a #/25 and how many times you've completed it. Most people say that this does not unlock even though most have completed it. To date I have completed it twice and am at 23/25 towards completion number 3.
Edit 2: waited and made sure to unlock the 25th successful mission via the multiplayer hub in the game, not the app. As soon as I did the debrief, I got the trophy. It would seem that using the app to get the successful missions is fine as long as the final (25th) mission debrief occurs in game, not in app.