Before instilling eye drops, you should wash your hands with soaps first. Then you can ask your child to tilt his or her eyes, and then warm the eye drops in your hand for few minutes if your child is too young for the warm eye drops will not make him or her blink so much. You can use your fingers to open his or her eyes to instill eye drops.
I think we should not to open child's eyes with fingers. You can ask the child to lie down and then place the bottle of the eye drops above his or her eyes and then ask him not move and look at the bottle. Then you can press the bottle when his or her eyes are open. I think most of eye drops will flow out of his or her eyes. It is a common phenomenon and don't worry about that. The drops in his or her eyes are enough to cure an infection or something else.