All the achievement says is "Buyer's Remorse".
What everything I checked online says is "Clear a map on any difficulty using only the starter pistol".
So I went and got it, using what info I had, and what my friend also did, I can make a quick guide to help others know what to do.

Rule 1: Complete a map.
Rule 2: Any difficulty is permitted.
Rule 3: Only use your pistol.
Rule 4: No melee, no grenades, no other weapons. If you even swing your knife, you're disqualified. Picking up the first weapon that you are forced to obtain after room 1 doesn't disqualify you, but you can't fire it, so pick a primary and swap off it immediately.
Rule 5: Shop is permitted, but you can ONLY upgrade your starting pistol. So feel free to pick up scrap and tech, just not equipment like helmets or grenades.

EDIT: Apparently you can actually pick things up and not get disqualified, so armor upgrades are allowed. Thanks to Kobayashi Maru for additional testing.

Spend your tech, if you got any. You can bring whatever tech upgrades you want into the run. Secondary Damage and anything to boost defense or mobility will help in this particular case.

Choose a map. You can run any that you fancy on any difficulty. I recommend Uniform Chaos in zone 1 on Adventure, as it's the easiest and fastest with no big boss fight.

Run through the map, shoot things, pretty much don't interact with any equipment drops but vacuum up any scrap you want. When you get your first weapon cache, grab a primary. If you don't get an option, just reset the run and try again - you only need to run room 1 again.

When you get your first augment, playstyle is most important here. Since you can upgrade your pistol with scrap, you will get an element, but you can't really force any certain element, so if you want to take an elemental upgrade (or are forced to choose one), elemental build augments aren't useless. Personally, I like to ADS, so I prioritize Aiming: Weakpoint and Aiming: Critical, but the starter pistol is so versatile you can make anything work.

When you get your first perk, I recommend prioritizing gun damage. Don't get melee or grenades, obviously. Lead head (If unlocked) is really good here. Otherwise just make do. Even if you get Vitalizer, Companion Robot and Lightning with a Fire pistol, you should be just fine especially on adventure, so just play safer. (Note: I don't know if companion robot damage invalidates the run. I would test it if I could.)

For items, focus on things that benefit your playstyle first, like weakpoint damage for a weakpoint augment, then more damage, then more mobility, then survival. Heavily consider epic augments if you see them. Avoid damage reflection (because I also did not test it in my successful run). Use your judgement.

When you hit checkpoints, feel free to splurge on your pistol and move on. In my successful run I didn't upgrade my pistol, and my friend did, so that's why I recommend it.

The pistol is semi-automatic. So you can hold down the fire button for automatic fire, but spamming the trigger fires faster.

In most situations this isn't too important. One upgrade for fire rate should come close to if not equal to spam clicking, and has less effect on your wrist. But I've heard someone bind their fire button to mouse wheel before.