There are several children in the cities that can be adopted without killing Grelod. One in Whiterun, one in Dawnstar, one in Windhelm, and one other that I don't remeber off-hand.
To adopt from Honorhall, however, requires the following and ALL ARE MANDANTORY:
Kill Grelod.
Talk to the Arentino kid in Windhelm (this also starts the DB quest, btw).
Have a place for the kids to sleep (there is an option for all the city houses, and Hearthfire main halls and Bedroom wings are automatic).
Have a place for them to store their "treasures" (City houses are complete with this, fully built Hearthfire Main halls also, Bedroom wings need the Child's chests).
Best way to know where you have these things is with your in-game spouse. He/she will only move somewhere where the above has been done.
EDIT: you can only adopt up to 2 children in a vanilla (unmodded) game. period.