As littlejay2o02 said, you need the Family Pack from the Road to Rule to be able to adopt a child.
What you don't have to be though is king/queen. You don't need to be anyone "special", as soon as you get Bowerstone Industrial, you can reach the Orpahange. But, as the others said, if you become king/queen and chose brothel instead of Orphanage, your chances to adopt will be lower.
The easiest way to reach the Orpahange is to fast travel to Bowerstone Industrial. When you get there, you see a bridge before you. Go straight ahead till you see a second bridge, not far from the one mentioned. Go through the bridge (to the left side) and you will see some beggars sitting under a fence. That fence is the Orpahange's fence. The building is distinctive, because there is a rather large (and dark) yard in front of it and its door is unusually big.
A red-headed woman (named Linda if I recall correctly?, quest called Kidnapped) with some folks is standing there with a 30 seal quest waiting for you, if you haven't done it before. If it's available, then it could be a useful fast-travel right from the map - easy Orphanage, tada!
When you're inside, go straight ahead, don't bother with the first doors. At the end of the hall lies the room you need - it's on the right with children's beds. Pick your favorite child and adopt him/her. You can adopt as many as you like, after the procedure, you have to select a home for them and a nanny will be instantly hired to take care of the kids. You can also have multiple children in one house (my selected houses only allowed two/house though).