Do not let the significant gamerscore or name of the mode throw you off. Challenge Mode is very easy and is nowhere near as difficult as the New York Minute Modes.
In the Main Menu, go to the Arcade Menu and you will find Challenge Mode. In this mode, you will have to complete the Story while a game-pausing challenge will appear on the screen before major firefights. These challenges are quite fun, and not too difficult. Some of the harder challenges include, "do not use cover" , "no headshots" , "fire only 10 rounds" , "no pills."
If you happen to fail a challenge, do not worry. You will restart from the last checkpoint you earned, which is always right before one of the challenges. Make your way through each level and the achievement will unlock once you blow up Rodrigo's plane on the last Chapter.