These are both located in the multiplayer menu of the game. Alternatively they can be accessed from the strike team mission console after speaking to Kandros on the Nexus. There is also a strike team mission terminal on the Tempest. From there you can go to "strike teams." You have 1 strike team to start. You can recruit more teams using mission points. Your first additional team will cost 40, and it seems to go up by 40 more mission points for each team you recruit. 1st team is 40, 2nd team is 80, 3rd is 120. I only have the 3 teams, so I don't know if that trend continues from there. Anyway, when in the strike teams menu you can select missions to send your teams out on. Some of these missions will have the Apex logo next to them. You will have the choice to send a team, or play in multiplayer for additional rewards upon successful completion. Successfully complete 25 and the achievement is yours.
You can also send strike teams using the Apex HQ app on iOS and Android. ***WARNING*** I'm completely unsure how the app communicates with the Xbox as far as achievements. I imagine you have to at least login to your game for the achievements to pop, cuz they surely won't pop in the app. But I am unaware if you get no credit whatsoever for using the app. I unlocked the achievement prior the apps release.
It's looking as though completing missions within the app aren't being tracked for the achievement progression at this time.
Confirmation that someone got this achievement from using the app first, then accessing the strike team menus in game. Achievement having popped after about 30 seconds of accessing appropriate in game menus.
A lot of people are having issues with the progression of this achievement tracking properly. Some people in the comments mentioned different things that seemed to cause the game to "catch up" with progression, if you will. From anything to recruiting another strike team, to actually playing an APEX mission in multiplayer. Keep in mind, just cuz it worked for others, doesn't mean it will be the fix for you. Just another thing worth trying. Unfortunately it just seems that the tracking on this achievement is pretty buggy at the moment.
Courtesy of a few commenters it looks as though checking your status via the prestige section and making sure to debrief number 25, and then access the prestige page again to have it start back at 0/25 is making the achievement pop. Could be another random way to force progression, but multiple people are reporting this method works.