For this achievement player must complete 5 APEX missions.
APEX missions are a subset of strike team missions. Not all strike team missions are APEX missions. APEX missions are indicated by Andromeda symbol on them (icon). These missions - unlike regular strike team missions - can be played online OR player can send strike teams to complete them.
Achievement requiers to complete 5 of those missions - win or lose. You can't combine methods of obtaining it though, you have to choose in which method are you going to get five games - are you going to play them online in multiplayer, or are you going to send strike teams to complete them (you can't play 3 online and send 2 strike teams to unlock this achievement - have to be 5 in selected route).
Progress can be checked in game's multiplayer menu, on challenges section.
Upon completion player is rewarded with stuff for single player campaign and points to unlock more strike teams or equipment for teams.
Player can use APEX HQ mobile app (EA profile tied to gamertag is needed along wit mobile of course :) ) in order to make progress towards this achievement. That's the way I did it - on xbox one I sent 2 or 3 teams, rest was done on mobile.
A lot of people have trouble unlocking this achievement. I too was frustrated that the achievement didn't unlock for me when it should. The way I tried to do it was from Tempest deck in single player game. Strike missions can also be accessed from multiplayer menu. So I went there, started to browse through menus and when I opened APEX challenges section both achievements - this and Veteran - unlocked.
Note, that for me progress bar for this achievement on xbox one was at the 0% all the time - it never went up. Other gamers reported that this is also the case, so you shouldn't stress too much about it.
My guess is that you have to complete tutorial in order to start making progress on it / unlock it.
So you don't have to accept missions results on strike missions screen in order to unlock this achievement. It may be needed though because you have to trigger something in game menus which will check mission counter in order to unlock achievement (eg browse through APEX challenges in multiplayer menu).
Also, my suggestion is not to mix two methods of obtaining this achievement, as this may be a factor (eg: when you switch method, achievement counter gets reset to 0). But it may be a wild guess.