I spend a lot of time at the airport in Mexico City and I always struggled to connect to the internet. There is WiFi offered through Telmex (Mexico's main telecommunications provider), but it is not free to use unless you are a client of the Telmex Prodigy Infinitum internet service. Telmex offers a free 15 minute trial which you may connect to by providing your name, phone number and e-mail address. This is great if you just want to quickly check your e-mail or let someone at your destination know that your flight has been delayed, but if you would like to connect for longer (and like me you want to actually get some work done), you will need to find another method.
If you have Skype you can connect to the internet at the airport with the Boingo service. The cost for this service is $0.19 USD per minute at present. You connect to the "Infinitum Movil" account and if you're signed in to your Skype account you will automatically get a message that you can connect to the internet through Boingo. If you don't have credit on your Skype account you will be out of luck in this case. If you use the internet for any length of time you will see that your Skype credit is quickly whittled away. Boingo also offers subscription options, which may be a good idea for you if you're a frequent traveler. More info from the Boingo website.
On one occasion when I was delayed at the airport for several hours I ended up spending quite a bit more than I had anticipated by paying for WiFi through the Boingo service. Ironically I had been feeling quite righteous because I had resisted my initial urge to hit the Starbucks for a calorie-rich drink, and was sipping a bottle of water while paying through the nose for my internet service. I later realized that if I had bought something at Starbucks, I would have received sign-in information to connect to the internet for free. Lesson learned! Next time I have a lay-over in Mexico City I won't hesitate to give in to the urge to go to Starbucks first.
When making your purchase at Starbucks, be sure to ask for the receipt for your transaction. The login info is on the printed receipt. You connect to the "Infinitum Movil" signal and then put in the user name and password that are listed on your Starbucks receipt. Then enjoy free WiFi from anywhere in the airport for as long as you like!
Extra tip: order a vanilla rooibos latte and take it without sweetener. The smallest size will set you back about 45 pesos (yes everything is more expensive at the airport), and you won't have to feel guilty for blowing your diet.
Read more about the Mexico City airport.