There are several ways to unlock this achievement, but the fastest and simplest way I could figure out is to attempt this during a Scavenger Run side mission. Scavenger Runs require you to pick up three loot crates before reaching the safe zone and do not require you to defeat any enemies. I unlocked this during Act 2, Chapter 8 on Insane difficulty, once I had a pool of mods that boosted my mobility.
You will want to use a Scout for this solo run, as they have skills that enable high mobility. Cloak is a valuable skill for this (your starting skill), as is Sprint LV3 (located in the Recon branch). The passive Free Cloak (giving you 1 AP for using Cloak; far right of the skill tree) and Quick Cloak (reduces Cloak cooldown by 1 turn; far left of the skill tree) are helpful, but not completely necessary if you don't have enough skill points to unlock them all. Having a point in Exertion LV1 (Raider tree) can also push you across the finish line. My Scout was level 4 and only had Cloak and Sprint LV1, so you can still navigate a battlefield despite un-optimized skills with some luck.
The trickiest part of this run is having mods that boost your movement. Mod drops are generally random, so you'll have to rely on the state of your armory to find armor that will assist your movement or AP. Chest armor like the Ranger Kit can give you a bonus action if you do not take a shot on the previous turn. Lower armor like Ranger Treads can give you increased movement. Even certain weapon mods can increase movement or AP restoration, if you're lucky.
When you're ready, select the single Scout and select the Scavenger Run. While running the map, chart out your planned route for the three easiest loot chests to pick up, and make a beeline for them, avoiding fights when possible. If you have a long, straight path with no obstructions, you can Sprint to cover a long distance with a single AP. If there are overwatch zones or wretches in your path, use Cloak to pass through them undetected. NOTE: If you pick up a loot chest while you're cloaked, it will count as an action and de-cloak you. Don't let this happen if you're ending your turn near enemies!
While you're running, you'll receive checkpoints and enemies will continue to spawn in your vicinity. From my testing, aside from E-Hole spawns, enemies cannot spawn and attack you on the same turn. Sometimes they are a bad spawn that will get in your way, so if this happens, reload the checkpoint and see if you can manipulate the spawns somewhere less intrusive. This method takes some luck, but eventually you can get a favorable spawn. If it seems like there's no chance to navigate a particular spawn, you unfortunately may need to restart the mission instead of the checkpoint.
Once you've collected the final loot chest, the evac zone will appear. As long as you reach the circle, the mission will end without the enemy taking another turn and you will receive the achievement once you progress past the rewards screen.
BONUS: If it's possible, you can try to shoot one enemy with a Snub Pistol then complete the mission to unlock the High Noon achievement. I did not get an opportunity to do this on my run, but if you have an extra AP before reaching the evac zone, it could be a good opportunity.
WARNING: This achievement does not work with the Rescue side mission. That side mission is very misleading, as the deploy requirements say you need a max of one squad member, but you end up saving two recruits during the mission. The game treats your deployment as three squad members at the end of the mission, which invalidates the achievement requirement.