This is a fairly easy achievement that you can get in about 30 minutes to 1 hour.
To start, purchase a tranquilizer rifle and an echo-lure from The Business in Fortuna. There are about 3 or 4 different animals you can capture, but for the sake of the guide buy the Pobbers lure. You can get both the gun and lure for as little as 1,000 Solaris United standing. When you have these, put the tranq rifle in your gear wheel. Also, make sure you DO NOT have a pet equipped when you do this, as they can scare the animals away.
When you get to Orb Vallis and equip the rifle, little paw print icons will appear on your map. When you go to them, you’ll come to some animal poop that you can “activate” (yum!), which then spawns some footprints for you to follow. Follow them slowly, as they can sort of sometime be difficult to see.
Once you followed the prints to the lure location, make sure no enemies are around, then take out the echo-lure where you’ll be prompted to partake in a little mini game of sorts. Activate the lure once, and on your second use you’ll see a white line. All you have to do is keep the lure line fairly close to the white one so you’re mimicking the animal’s call. It’s a lot easier than it sounds. You’ll know you’ve done it correctly when Business tells you the animals are on their way.
Next, you want to hide somewhere so they don’t see you! I recommend a high vantage point. The Pobbers are small, possum-like creatures who can spawn in groups of 1-3. If you’re having trouble spotting them, looking through the scope of the tranq rifle will light them up, even through walls/trees. When they stop moving, shoot one with your tranq rifle. The shot will scare the other ones, so just go grab the one that was unaware for your “perfect” rated capture. The scared ones are not worth getting as they’ll only get you a “good” rating.
Hope the guide helps you. If you are unsure of any specific step, feel free to ask in the comments. Good luck, Tenno!
Edit: Due to a patch, sleeping and cloaking abilities no longer work for this achievement, so I removed them from my solution. Still, with this strategy you should be able to do it quickly.