There are two ways about obtaining this achievement. The quick and expensive way is to simply pay for the entire collection at the marketplace and the achievement will pop. The slow but cheap way is to keep playing games earning booster packs until you unlock all the cards.
On each of the 5 worlds there are 4-5 enemy battles and one "explore" battle icon. None of these explore icons are unlocked until you beat avacyn in the innistrad section. Each of these explore icons will give you a random battle with a different theme deck tied to that world. there are 5 different battles you can fight in each icon and none of them will repeat until you have defeated all 5 at least once. After you have finished each battle in the same node (each time earning a booster pack) you can continue to fight these 5 decks in a random order earning more booster packs until you have completed that worlds collection. You can continue fighting the normal story battles too but only the explore icons will net you additional booster packs after the first win. When you have completed one world move on to the next and so on. Each booster pack will net you about 10-14 cards. 7-11 commons, 3 uncommons, and 1 rare/mythic.
In the card collection page it will tell you at the bottom how many you have collected. You can hold 4 of each common, 3 of each uncommon, 2 of each rare and only 1 copy of a mythic rare. Cards you do not own are greyed out and cards you do own are full color with a white and black number icon on the bottom left of the card image. This image will turn orange and black when you have maxed out the number of copies you can have of that card.
You can not own more copies of a card than its rarity allows. For example, once you open your 3rd Hedron crab (an uncommon) it becomes locked in packs and you can not obtain any more. As long as you keep earning packs, you will eventually earn everything.
It has been reported that a few have unlocked this achievement early despite not having the entire collection yet. If it is glitched it seems to glitch early rather than late/never. For myself it did not unlock until the final card collected.
One final note: you do NOT have to purchase any premium boosters from the store or complete the premium collection page to get this achievement. You only need to 100% the collection pages for Innistrad, Theros, Ravnica, Shandalar, and Zendikar.
edit: You get 4 bonus rares in a booster pack for defeating the boss battle in each node (Chandra in Ravnica for example) you can earn these as many times as the boss battle shows up. Credit goes to Meiphora for discovering this.