Is my refusal to perform a service for my colleague because I dislike him considered backbiting, and what should I do to avoid being thought ill of, especially if the person requesting the service is a parent?

The reality of backbiting (gheebah) is when a Muslim mentions his brother with something bad in his absence, as the Prophet, peace and blessings be upon him, explained it by saying: "Your mentioning your brother with something he dislikes." Merely stating that one does not like a person is not backbiting, unless it involves denigration or describing them with something bad.