In order to do this much easier, I recommend you to get the Houdini plasmid (lvl 16) first.

When an enemy becomes a Big Daddy, find him but dont alert him. First make sure you have full Eve.

If he has his back turned, keep using the melee attack on him. When he starts to turn around, use the Houdini plasmid to go behind him again and melee.

Keep doing this until you either run out of Eve, die or kill him.

The melee attack does more damage than any weapon or plasmid.