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
In order for a soldier to rank up, he/she has to kill a set number of aliens. I do not know if it has to all be on one mission, but anecdotal evidence through my playthough indicates that it is kills in total, not kills in one mission, and that which type of alien is killed doesn't matter. My Colonel (the highest rank) with the least number of kills has 24 kills across all missions. My Major (second highest rank) with the most kills has 18 across all missions. I believe that recruit to squaddie is 1 kill, and squaddie to Corporal is 3 kills. Not definite about any others.