This achievement will be much easier with a dedicated group as you can split up and get the mission done faster. Make sure to select bugs as they are easier, and don't worry about any sub-objectives, just destroy the required number of holes as fast as possible. To clarify, this is the "Blitz: Search and Destroy" mission type. There are many ways to go about this, but I will share how I did. It is one of the harder achievements so I would save it for last when you have more equipment unlocked.
If you are doing this solo like me, you will not have enough grenades to do this in 6 minutes. For stratagems, I would equip the grenade launcher (unlocked at Level 5) and the EXO-49 Emancipator Exosuit (unlocked at Level 25). This is because you can use these as weapons to close all of the holes instead of simply relying on grenades. It's doable with the grenade launcher only, but the suit allows you to steamroll through tons of enemies that would usually slow you down quite a lot, and time isn't on your side here. If you're just starting, level 25 may seem like a while away, but by the time you even get close to finishing the 100 missions' achievement you will be well past level 25. You could also equip the Jetpack stratagem in order to get around faster, as well as throwing your stratagems far in front of you while you run in order to reduce idle time waiting for your stratagem to land.
You only need to close 9-12 holes for the mission depending on your luck, so I would drop at the biggest nest in order to get them all done without having to travel around the map a ton. Lastly, I would try and search for a mission that has the extraction point close to a large amount of bug holes to reduce travel time. This is completely randomized but can really make the difference.
All of this applies to doing it with a group of people as well but is less important as you can split up and have enough grenades between everyone to get all of the holes required. However, if you do the mission solo extraction will only take 2 minutes instead of 3, so it is a tradeoff.