Despite the rules saying that ties are won by the player with the most face cards, this is not how it actually works. You will lose every tie. This is an important thing to note, because it means you will not win approximately 1/3 matches. Random playing will, over time, lose you money. You have to fold to increase your chances of winning in the long run.
Therefore: fold. Often.
If you are dealt anything lower than an 8 on your first card, fold. You now have a below average card and a significantly increased chance of failing to make a good hand. Even folding 8s is a good move, since that is only an average card. And by average here, I mean the value of the cards in this game averaged across all cards. Frequency-wise, obviously 10s are the most common.
If you are dealt anything lower than an 8 on your second card, check it against the revealed card. If it beat both the opponents you can consider playing it, but it's probably safer to fold. If your low card is lower than their low card (or worse their face card) fold.
After your third card check the positions again. If either player getting a 10 will beat or tie your score, it's likely the correct move to fold. It's not enough to beat one player, you have to beat them both. Folding here can be painful because you might feel pot committed, but saving 3 gold is still worthwhile.
The earlier you fold, the more money you save. The default positions here are losing 3 gold or winning 24. If you start to play a hand and miss, you significantly increase the frequency with which you need to win. Instead of once every eight hands to break even, it's now once every 7, or 6. And we already know because of the ties situation that our win rate, unless we are lucky, will be less than 1 every 3. The margins start to get small unless you aggressively fold bad hands.
Getting 3 of the face cards and/or aces usually wins, but not always. I've had 30 lose to 31 too often. A hand that scores 30+ happens about 5% of the time. Getting a 2-7 on your first card is another 46% of the hands. Folding those means your already folding almost half your hands. Figuring out which of the remaining 49% of the hands you're dealt to play and which to fold is where you'll earn this achievement or not.
The achievement pops as soon as you are +100 gold, you do not have to leave the table with the profit. Therefore, if you're hoping for luck, you can just turbo "A" to keep playing and maybe eventually get lucky. Even though the long-term prognosis for that is bad, maybe you get a few lucky hands in and earn the achievement anyway. The game is very high variance. The AIs regularly trade back and forth to +/- 200 coins.
Good luck, you'll need it, because this achievement is all about the luck.