Question: Soldiers gain experience from killing enemies. At what experience levels do the soldiers advance to the next rank?

Answer: 
The XP chart for promotions is correct. However - you don't have to kill to earn points. For example: A Sectoid kill gives you 30pts. A successful abduction mission gives you 80pts. 

Make 1 Kill, survive, accomplish mission = 110pts aka Squadie promotion.
Don't make a kill, survive 2 accomplished missions = 160pts
I had a colonel after 29 kills that just hadn't been on that many missions, she was a psi and i had a large dry spell of psi soldiers. My primary goal in every mission for my team was to get her as many kills as possible as quickly as possible. I have also had colonels early game with much fewer kills (high teens i think is my lowest...mebe 19...not verified) simply because they were along for the ride as my overwatched squadsight sniper got 1/3+ of all kills. I have definitely had soldiers level up from mission completes without getting a kill.