To unlock Hive missions, you must first advance the "Reigning Supreme" Mission Chain to Step 2.
I found this guide already created for how to complete each task in part 1: https://www.ign.com/wikis/marvels-avengers/The_Days_Ahead#Th...
The first Hive mission you will encounter is "Let The Game Begin".
After completing this mission, other Hive missions will be accessible from the War Table.
Alternatively, each mission that is part of the weekly Mega Hive mission chain counts towards this achievement as well. The downside to doing Mega Hives is that you will be playing solo, and if downed at any point, you will fail the mission (and the mission chain restarts).
Each Hive mission consists of around 6 or 7 floors of an AIM facility. Each floor will have a different objective (usually defeating enemies, destroying servers, controlling zones, protecting defecting AIM scientists, etc). Each mission usually takes 15-20 minutes or so. After 5 hive missions (including any mega hive missions), this achievement should pop.