To be eligible to adopt a child, you must acquire either a house with a child's bedroom or a homestead furnished with a bed and a chest. This solution will follow the former method of purchasing a house.
This achievement can be easily obtained in Whiterun. Upon completion of the
"Bleak Falls Barrow" quest you are given the opportunity to purchase Breezehome from the Jarl's assistant,
Proventus Aveniccia. After purchasing the house, Proventus will also sell you furnishings for it including the child's bedroom for 500 gold which you must purchase for this achievement.
Breezehome will cost 5,000 gold and is located next to the Blacksmith.
After acquiring Breezehome and furnishing it with a child's room, head to the big tree in Whiterun. Sitting on a bench by the big tree is
Lucia, a young girl who is begging for gold. Give her a gold piece and ask her why she's begging, after which you will get the option to adopt her. Do so and head back to Breezehome where she will be waiting for you.
Talk to her again and the achievement should pop.
Also keep in mind that Breezehome has enough beds for 2 children so you may adopt another child in addition to Lucia, if you wish, but only one child is needed for the achievement.
Here are a few more children who are also available for adoption that can be found in other cities:
Alesan – In Dawnstar near the mines (Boy).
Blaise – Just outside Solitude at Katla's Farm (Boy).
Sofie – In Windhelm, Gray Quarter (Girl).
Also by giving a gift or gold (any amount) to your adopted child you will gain
The Gift of Charity which will increase Speech by 10 points for one hour.
Good luck and enjoy being a parent
.