Is what this man did considered slandering chaste women (qadhf al-muhsanat), or backbiting (gheebah), taking into account that he said it based on what a woman told him, and that he is in enmity with the woman's family, and how can he expiate for his sin?

A man's statement about a woman, "She looks at him with bad gazes," is not a slander that necessitates a prescribed punishment (hadd). Rather, it is backbiting (gheebah) that is forbidden, except in specific circumstances, such as aiding in changing a reprehensible act. If a man mentions this without a valid interest, he must repent to Allah, speak well of the woman in front of those before whom he spoke ill of her, provided no harm results from this, and seek forgiveness for her. This is sufficient without informing her.