Being completely alone, with no one else present — emphasizes solitude or being by oneself.
週末は家で一人きりで静かに過ごしたい。
I want to spend the weekend quietly at home all by myself.
彼は仕事の都合で一人きりで海外に行った。
Due to work circumstances he went abroad alone.
子どもが夜中に一人きりで泣くと心配だ。
It's worrisome if a child cries alone in the middle of the night.