I did the quest for the little boy in Windhelm. Now I'm just sitting in Windhelm, and sometimes walking out, and just waiting for random hours for a courier to come and he still hasn't. Can I not be around Windhelm, or I am not waiting long enough? I've waited about three or four days in total...

Mine took a few days to arrive. Be patient. It'll eventually come

Go to different cities and do stuff instead of just waiting. The courier found me a few days later when I was turning in a quest in whiterun

He's usually waiting for you in another city or town.

They came to me when I was traveling to the next town on foot.

Maybe that's what you need to do to set it in motion.

Or just try traveling anywhere with fast travel. Like to a small town with not many guards.

Do I need to be past a certain point in the game? Because I'm level two and just got out of Helgen. I made this guy just to do the Dark Brotherhood, and pick him back up later as my assassin type character.

After fulfilling the boy's request, going to sleep in an inn advanced the dark brotherhood quest for me.

He comes at random, on one of my characters he came immediately after I told the boy she was dead, on another it took a few days and he found me in Riverwood. Make sure you did go back to tell the boy she was dead, though. My first time I forgot that part and was wondering why no courier ever showed up.