DPS Idle

Method:

Updated. I can't comment because TA won't let me, but thank you for the feedback, I have incorporated it below :)

Reports have been coming in that the achievement will also unlock when you hire the first hero, the Hunter. You will need to "silver" the first deck eight times to do this (no small feat when the multiplier is so broken).

Below I recommend the first deck, as some decks (like the daily login one) don't unlock the achievement when you get the four boosts required on a card.

VERY random.

On the first deck, quickly skip past the hog (second enemy) as soon as you can. Keep playing the game as normal, saving up your leaves, until you get the opportunity to turn a card silver. Along the top row of cards are a couple that let you increase the chances of getting leaves from the enemies which drop them - turn THIS card silver. Now, go back to fighting the hog and watch the leaves roll in!

Gather up all the leaves you can, and click on the small number above the "boost" button on the right hand side of the screen until it shows "99" (purely for convenience). Then start hitting the boost button itself! This will add small boosts to a random card's stats from your current deck, as signalled by a small sword icon in the corner of a card. Each card can have several of these.

Keep on clicking the boost until the RNG gives you four boosts on one card, and that's it! Achievement unlocked. Again though, this is VERY random and there's no way to influence where the boost goes.

This achievement can unlock even if you haven't gotten the card yet, as the boost can be applied to cards you haven't earned. So just click away until you get it.

The above method IS what you need to do. That is the solution. Downvote the RNG, not me ^_^

Small technique for those who have not yet launched DPS Idle or who have launched it, but you cannot manage to have the achievement of boosting a card 4 times, because this is based on luck so once you have reached minimum 312 leafs you boost if you had not done it you do Xbox button quit the game does not make you directly Manage the add-ons you delete your save from the console not everywhere this will make you quit the game automatically and when you go on the games you always had your 312 leafs repeat the manipulation I do it 5 times and after the 6th time it works.

Well, actually this way worked for me, and has nothing to do with the ones above or the description itself.
For unlocking this I have to have 8 silver cards on the first set, which would unlock the first hero.
At that very exact moment the achievement unlocked for me, on the Windows version too.
Being capable to unlock just 8 silver cards upgrading them get you this.
Perhaps the other methods above work too but, I was not able to replicate, due RNG and leaf number not enough to do it, and fall into the right cards. You don't need luck for this.

Just trying to help here, not saying that the other methods do not work, perhaps this game is so cheap that they did not checked and could be patched in the future.

1. Try to get a card to level 3 as early as you can.
2. Get a lot of leaves (+100) by killing the boar or dendroid, (etc.).
3. When you are in a good state, shut the game down, so the current save uploads to cloud, so it is stored in Xbox Live.
4. Start the game again. Now, press the boost-button and use all the leaves you can and try to get the level 3 card to level 4.
5. If it fails, press the Xbox-button, go to Manage game and add-ons, go to Saved data and delete it (only local data, not everywhere, if that is an option)
6. Now start the game again. The cloud save restores your leaves. Try to boost the card to level 4 again.
7. Keep repeating steps 4 to 6 until the achievement unlocks.