This is a quick (mostly) spoiler free guide on how to find the "dog" and get the "Best Friend" hidden achievement. All spoiler images are later in the guide. For a truly spoiler free experience, just read the TLDR. For a more in-depth explanation, read on.

The steps to complete this achievement are simple on the surface.
1. Search levels for 'Hidden Bunkers' until you find one with a "dog" inside of it.
2. Rescue the "dog" by picking it up just like any other ammo drop, health kit, etc.
3. The "dog" will occupy one of your consumable slots until the end of the run.
4. Beat the run.

For a more detailed explanation of factors that probably do and don't matter for finding the "dog", as well as a 'How to' on easily finding "Hidden Bunkers", see below.

Spoilers Below

If you've played more than a few runs you've more than likely accidentally stumbled upon a "hidden bunker" loot room in a level at some point. While these might initially seem random, they can be reliably and easily found in around half of all levels.















There are visual clues in levels for if, and where, these secret rooms are hidden. The easiest way to find one is to look for this small ladder symbol that can (usually) be found in several different locations in a level, but only if a secret bunker has spawned at all.
This ladder symbol always has a small arrow as part of it's design that indicates the rough direction of the secret bunker.
















Continue following the directions of these arrows (could be all the way across the level) until you spot a crack in the ground like this one. Now simply use any explosion (red barrel, grenade, enemy explosion, explosive weapon, weapon mod, etc.) to reveal the entrance to the "Hidden Bunker".

You will need to keep farming "Hidden Bunkers" (possibly only on the first level) until you get lucky enough to find a "dog" roaming around inside. In my case it was crawling on the chest in the middle of the room when I first entered. You will also notice a dog bed, dog bowl, and several suspiciously human-shaped bones it has been gnawing on.

You will need to pick up the "dog" like you would any other piece of consumable equipment in game, and then come to the horrifying realization that this little scamp will be occupying one of your two available equipment spots for the rest of the run.

Hopefully, you don't find it during a heat 15 solo run like I did. At this point you need to keep it in your inventory until you beat your run. Be VERY careful to not drop the poor thing at any point. If you try to pick up a second consumable, you may drop them (it can be picked back up)

These are the things that probably affect successfully completing this achievement, though I completed the achievement on my first run where I encountered the "dog", so I have not tested for fail states.

1. It seems the "dog" only appears in a secret bunker on the first level. It may be worth resetting the run if you don't find it right away.
2. Keeping the "dog" in your inventory at the end of each level.
3. Beating the run boss with the "dog" still in your inventory.
4. Every instance reported so far of the "dog" appearing has been in solo mode

At this point it is unknown if it is possible to find the "dog" in multiplayer, and if it can be found, it is unknown if it counts for everyone, or only the player who carries it.

Hopefully you were able to find your new companion without too much headache. You better not forget to feed it.

In case anyone is curious, even after you save the "dog", other ones can be found, but rescuing them will not add extra "dogs" to your bunker (my disappointment was immeasurable).