The more decisions to make in the game, the better - focus on the risk
Describe each decision the character makes in detail. What is the most useful move your character could make in this moment? Why is it the most useful one? What are the assumptions behind this? How would her opponent have to react to make this useful? Will it instead hurt her if her opponent calls her bluff?
This all boils down to: describe the risk involved in each turn.
For example you could start with a few basic turns that are basically standard in the game you chose. A traditional opening in chess for example. But then the opponent makes a turn that normally doesn't make sense.
Why would he do that? What is his strategy? What am I overlooking?
Later your character has to decide: If I do this I will likely gain an advantage. Except for when he interrupts with a certain other figure. That would give me a momentary disadvantage, but in the long run the positioning of my other pieces is better.
Describe the possible consequences in the character's life
And make her think about the consequences of the game itself. How much will she lose? How will this affect her? Will she be so poor that she is not sure whether she will survive? Is there an actual physical cost involved, such as playing Russian Roulette (probably not an interesting game as there is not a lot of strategy involved) or will someone she loves be affected by the result because they lent her a lot of money?
Make every point count
How does it feel for your character to lose a point/piece/round/...? How does she react to the rising fear of losing the game? How does she feel about winning a point?
How are the players interacting with each other - and the audience
How does she react when everything seems to be going according to plan? Does she taunt the enemy, only to realize that she overlooked an important piece on the board? What are they talking about? Is her opponent trying to distract her? Is he playing with her fears?
Is there a time limit that adds to the suspension?
What does the audience do? Are they silent? Loud? Who is their favourite? How are they reacting when their favourite loses/gains a point?
Inspiration from Anime
There are quite a few sources you could look at to get an idea of how this could be done. For example there are a few Animes that I have seen where the characters are basically only or at least often playing games of some sort.
Code Geass for example uses chess to show how the character is always trying to be ahead of his enemy. There are many instances where they are literally playing the game and many times where it's only mentioned as a metaphor. (At least as far as I remember; it's been a while since I've seen it.) The genre is basically mecha-fighting and quite a bit of magic. It's heavy on the strategy side and about playing people against each other.
Kakegurui is about a school where gambling is everything. Most people there are incredibly rich and the better you are at gambling, the more privileges you have. If you have a debt you are considered a pet for example. The main character (and quite a few others) are pretty crazy when it comes to gambling. They are betting their finger nails or even lifes on games and the possible strategies are often alluded to. Cheating is also an important part. The moments when someone realizes they lost are described in a good way and it's shown beforehand and afterwards what the implications of losing are so that the viewer knows that losing is not only to lose a bit of money. A little warning: the design of the female characters can be a bit... overdone if you know what I mean.
No Game No Life is another one in this regard. Two people, brother and sister, are transported into another dimension where literally anything can be decided by gambling and gambling is binding. Great for them as he is a genious when it comes to analyzing human behaviour and therefore bluffing/ seeing through bluffs and she is a genious when it comes to strategy. Perfect team. The beginning is quite good as they for example explain how the rules favour certain results and why they could simply go with possibilities to make a good starting in their new lifes. It's good to show how rules of unknown games can be analyzed to check for loopholes and problems and how knowing more about the game and its rules helps in getting the upper hand. But it quickly starts to lose this aspect and games become ever more random and the characters ever more overpowered. Bigger warning: these characters are definitely overdrawn and I would recommend watching maybe the first one or two episodes. I am pretty sure the first episode included one or two card games. The rest is probably not useful.
Gyakkyou Burai Kaiji: Ultimate Survivor is another one that I have never finished (I didn't like the artistic style). As far as I remember it featured a cruise where people had the chance to pay off their debts by participating in a tournament. Everyone got either rock, paper or scissors as a card in the beginning and you could ask anyone to play. The loser would lose a card and the winner got one. The goal was basically to be the last one standing. This one was mostly about the main character worrying about his normal life and about how he didn't understand the implications of the rules until it was too late and someone already got the upper hand.