Like basically all the achievements in the Falkreath DLC, "Proud Parent" is a cinch to earn and won't take you more than a few minutes.
First of all, you'll have to own a house. If you don't already have one, consult the TA guide for the achievement "Citizen" here:
Just hang around a major city for a while (i.e., the one you live in) and a courier will deliver you two letters, one called "Consider Adoption" (more on that later) and one from the steward of your hometown notifying you that you can now purchase a children's bedroom set for your home.
The option should be added to your Home Decorating Guide, or you can visit the steward himself and buy it from him. This will only cost you 250 gold. WARNING, buying the children's bedroom set will replace your alchemy lab (if you have one) and cost you any items that you had there.
(NOTE: User Belaric pointed out that the children's bedroom set costs 3,000 gold in Solitude in Windhelm and does not cost you any rooms. The steward will warn you if you are going to lose a room; he did for me when I bought the bedroom set for my Whiterun house.)
Once you buy the bedroom set, it's time to adopt! The "Consider Adoption" letter the courier delivered earlier will inform you that the orphans of Skyrim have been hit particularly hard by "these days of war and strife," and that you are encouraged to visit Honorhall Orphanage in Riften to adopt one of them.
Now all you have to do is head to Riften, visit the orphanage (in the southwestern corner of the local Riften map) and talk to a Constance Michel about adopting a child. She will ask you a few questions, such as "What do you do for a living?" If you answer wrong (like replying, "I'm an assassin") she'll reject you, but then she simply starts asking you the questions again and you can answer differently, so you can't really mess this up.
Then all you have to do is pick the little squirt that you want and this achievement will be yours! (I chose Runa Fair-Shield. She's a fiery spirit!)
This is my first guide, so please feel free to offer any feedback, suggestions or questions! (Note, I did not make the video below myself, but it should help you if you need further assistance.)