This one is a little confusing when you first see it. Paladins gives you ~200 cards automatically when you start the game, so when you go to check how many cards you have, you will think you already met the requirement. However, this achievement actually wants you to unlock
100 cards by yourself, so the ones you started with do not count toward it.
As of this writing, with Strix being the newest character, you start with 192 cards, so you'll need to unlock 100 beyond
that. Remember, this number goes up every time a new character is added because they give you free starter cards for every champion, so don't be surprised if the starting number is super high someday.
According to others, you can earn 36,000 Essence just from completing the tutorial
. I cannot confirm as I did it before the currency change happened. Credit to Eurydace in the comments section for that.
As for how to purchase cards...
Essence, the stuff on the right side in the purple bottle, can be used to directly purchase cards you want. If you don't have Essence, you can unlock random cards via Radiant Chests, which cost 2,500 gold each or 50 crystals. If a chest gives you a card you already have, the game will automatically give you Essence for your duplicate. You can then purchase the cards you want directly with Essence. They range from 500 to 12,000 Essence based on class/rarity, so go for the cheap ones if you're solely looking to get this achievement. You will also be refunded more or less Essence depending on the value of the duplicate you received. If you have a booster active at the time of opening chests, you can get 1 extra card per chest.
Select a champion and go to "Cards" to start purchasing cards for whatever character you want.
You can unlock more Radiant Chests by leveling up characters or your account, so watch out for Double XP weekends, and also Double Gold weekends to earn more gold to buy chests.