Are you considered a hypocrite if you associate with friends whose dealings with you you do not inwardly approve of, but you treat them well, especially if they do something like speak ill of someone and then try to befriend them?

Being kind to people and treating them well, even if one dislikes them, is not hypocrisy. Rather, it is a good deed intended to win their hearts and reform them. The Prophet, peace and blessings be upon him, was kind to those he disliked, and he said: "When have you known me to be obscene? Indeed, the worst of people in station with Allah on the Day of Resurrection is he whom people abandon out of fear of his evil."

Your companionship with people should be with the aim of reforming them or achieving a religious or worldly benefit. However, you should only befriend a believer who increases your closeness to Allah, as stated in the Hadith (definition): "Do not accompany anyone but a believer, and let no one eat your food except a pious person." It is also necessary to beware of backbiting and to interpret people's actions in the best possible light.