Which character from the short story “Rules of the Game” by Amy Tan is motivated by a desire to master “invisible strength” in life and in chess?

The character you're looking for is Waverly Jong, the first person narrator of the story. Her full name is actually Waverly Place Jong, named after the street she grew up on in San Francisco's Chinatown. The very first line of "Rules of The Game" comes from Waverly, who says:

I was six when my mother taught me the art of invisible strength. It was a strategy for winning arguments, respect from others, and eventually, though neither of us knew it at the time, chess games.

It turns out that Waverly's mother's advice not only applies to everyday life, but to chess as well. When Waverly's older brother Vincent is given a used chess set at their church's Christmas Party, the three Jong children become engrossed in the game, and Waverly begins studying the rules. As she learns more and more about the game by playing, she tells us:

I discovered that for the whole game one must gather invisible strengths and see the endgame before the game begins. I also found out why I should never reveal "why" to others. A little knowledge withheld is a great advantage one should store for future use. That is the power of chess. It is a game of secrets in which one must show and never tell.

Waverly eventually becomes something of a child chess prodigy, but this only seems to make her relationship with her mother and her own invisible strength more complicated.