The most difficult part of this achievement in my experience was that my shelter took a long time to get breached. I was around day 180 when it happened, and because it had taken so long, I was unprepared for when the attackers arrived. If this happens, or if you set up some traps and they fail, you can try dashboarding and closing the game before the day ends. The raid spawned again on the same day when I reloaded, and I had a couple extra hours to prepare traps for the intruders thanks to this trick.
In terms of traps, the best kind you can place are obviously the ones with the highest percentage, currently being the Shotgun Trap with an 80% hit rate. However, this uses a lot of harder to find materials, and if your breach spawns earlier than mine or you do not have the required items, Spike Traps (60% hit rate) may be a better option. The Shotgun Trap is in Tier 4 of the Workbench and the Spike Trap is in Tier 3, so if you do not have these workbench tiers, you will have to use lower hit rate traps.
Set up your traps in areas where the intruders are likely to walk. Right near the entrance works great, or just in front of an easily accessible pantry or storage box. You can build the traps prior to the raid to make sure you have enough time. I would recommend crafting at least 4 traps, even if they are of varying success rates, as I had 3 intruders in my breach and I'm unsure if there can be more. Even if you only get 1 or 2 invaders, it's better to be safe than sorry, and you can always deconstruct the traps after if necessary.
I would also recommend that you craft tripwires in advance of the breach. These are used to arm each of the traps, and so preparing them in advance will save you time once the shelter is under attack. You will need as many of them as you have traps, plus some shotgun ammo to arm the Shotgun Traps if you have any. You can only arm the traps when the breach has started, but you can craft the necessary items in advance.
When the breach does occur, arm all of your traps, and try to hide your shelter dwellers in Hiding Spots. If you don't have enough of these, direct them to stand as far away from the entrance as you can by mashing the
button to have them walk to your cursor. If your traps do fail, the invaders will likely kill at least one family member if they can see them. Of course, you could always dashboard and try to reload, but as I said earlier, I preferred to be careful. Having upgraded Shelter Doors gives you more time to arm the traps, as it delays the intruders as they try to break in.
Hopefully, your traps will take out all of the intruders, and your shelter will go back into normal mode. At this point your achievement should unlock. The achievement doesn't unlock if you get raided, but the intruders survive, so if it looks like this is going to occur, attempt to close the game and reload your save to the start of the day.
EDIT: iVilux suggested that if you have intruders come and damage/break one of the doors but then leave, do not fix the doors! This will hopefully stop the same thing from happening next time. I wasn't aware that this could happen, so maybe hold off on door upgrades if you think you'll have enough time to arm your traps anyway!