Update January 2020:
mentions in the comments:
This seems to have changed a bit since anyone's looked at this guide. Guardians are now available "a la carte" with a Crimson rotating each week and the others rotating daily, and you can see which one you'll get with no rng involved. Advise saving up for the crimson, especially if it's a good one like snake or ward, and then checking in daily to fill in the gaps. I believe the prices are 500 green 1000 blue 2500 purple 7500 gold 8000 crimson. Just waiting on that purple snake to show up...Note: After Patch 3.6 there's now 5 different tiers/colors for each guardian - a crimson red tier has been added and you need it to unlock the achievement (thanks Velodyne for the info). There was a pack available for a limited time in the Emporium for an Exclusive Crimson Bat (it cost 10k in Astral gems). The offer is no longer available, but might be back in the future, so it might be worth it to keep an eye on it just in case (thanks LL26xRoyst for the info)
To unlock this achievement, you need to collect one common, one rare, one epic and one killer guardian of the same type (that is, one of each color for either Vampire Bat, Fractured Ward, Ram, Snake or Exemplar).
Common guardians are green, Rares are blue, Epics are purple, and Killers are Gold.
You get guardians from packs you buy from Kan-Ra at the Emporium with in-game currency (astral gems) or real money (KI Gold). You have to buy packs until you get all 4 colors for the same guardian, which means you will get duplicates before having one of each color. The achievement pops when you open the pack with the color guardian you were missing.
After patch 3.6 (https://www.ultra-combo.com/3-6-patch-notes/
) you can hold 5 guardians of the same type at the same time. That is enough for this achievement, but since you'll probably get some duplicates before getting all 4 colors for the same type of guardian, you might need to destroy any excess guardians you get. So, let's say you have one Killer Ram, three Epic Rams and one Common Ram. You will need to destroy (transmute) any excess Rams you get (you get astral energy in exchange) until you get the blue one (Rare Ram) you need. So in this situation, make sure you destroy one of the Epic Rams (unless you get a Rare Ram, since the achievement pops when you open the pack, you'll then be free to transmute any Ram that you want).
The most efficient way to go around this is to buy first the Killer pack, which gives you one guaranteed Killer guardian, then move on to the Epic pack and purchase those until you get the Epic tier for the guardian whose Killer tier you already have, then move on to Rare packs until you get the Rare tier of your guardian, and then move on to the Common packs (if you still need to buy any guardian packs).
Fractured Ward is not the best choice of guardian to collect all tiers for, as I believe you can get those only when the special Forerunner packs are on sale. So if your only Killer guardian is the Fractured Ward, it's best to purchase a Killer pack and get a different one to work with.