It depends on their class. As far as I know, there's not a whole lot of evidence concerning how the lower classes relieved themselves because lower class dwellings are less often preserved. My educated guess would be that they relieved themselves in chamber pots and then emptied them in the Nile.

However, there is plenty of evidence that higher class people had rooms in their houses specifically for bathing and going to the bathroom. I know Katrina Szpakowska has some papers about the layout of Amarna mansions which mentions these "bathrooms". There have been stools with holes in the middle, interpreted as a kind of toilet, found in grave goods as well.