I would say that the French expression “je connais X sur le bout des doigts” is the closest to “I know X like the back of my hand” in terms of meaning and image. For meaning only, I agree with cl-r’s suggestions.

I would say connaître sur le bout des doigts, the others have slightly different meanings

• connaître par cœur is straightforward knowing by heart. See also apprendre par cœur → learn by rote
• connaître comme le fond de sa poche or just comme sa poche is generally use for place, e.g. one usually knows his hometown comme le fond de sa poche.