You push at it angling into the corner of the eye...put pressure downward so that the duct is between the ridge of the eye socket (bone) and your finger, then rock/push back and forth. It's not a sweeping motion at all--more like a hard press. The eye will sort of move to the side and out of place (a LITTLE...it's not as freaky as it sounds). It would really expell a lot of junk about once a day--sometimes twice. It was usually yellow and gooey, and I would have a wipe or kleenex handy to try to dab it out of the corner of his eye (otherwise it would just drain back toward the tear duct). Make sure your hands are clean when you do it--you don't want to introduce anything into the eye or the emptied duct. Doing it during nursing seemed to be the best. It might be bothersome to your baby, but it shouldn't hurt.
That, combined with breastmilk, combined with regular cleaning (I would get a wipe warm in my hand, then use it to get rid of the crust after every nap/overnight) and it worked. I think I did the pressure thing for a few weeks before it went away, when he was about 3 months old.
It WILL go away. I know it's tough to deal with it, but it will.
ETA...if you do a search about this, someone else mentioned this technique in an old post. She might have described it better than I did--but she said she was surprised at the pressure it took, too.