na-adjective, noun
strong-willed; determined; competitive
Describes a person, especially a woman or child, who is spirited, unyielding, and hates to lose. Often carries a positive nuance of lively determination, but can imply stubbornness depending on context.
彼女は勝ち気で、どんなことにも全力で取り組む。
She is strong-willed and puts her all into everything she does.
あの子は勝ち気だから、ゲームで
That child is so competitive that he cries when he loses a game.
負けず嫌い focuses on hating to lose and is often used for someone who is a sore loser or extremely competitive. 勝ち気 emphasizes a spirited, determined nature and is more about inner drive than reaction to losing.
気が強い is a broader expression for having a strong personality or being assertive. 勝ち気 specifically highlights a competitive, unyielding spirit.
Compound of 勝ち (kachi, 'winning') + 気 (ki, 'spirit, temperament'). The combination yields the meaning of a competitive, determined disposition.