To unlock this achievement, you need to shoot and detonate all of the explosive barrels located throughout each mission.
You can check your progress on the mission select screen. Highlight a mission and press "Y". If you have found all the barrels for that mission it will display the message "All Exploding Barrels Destroyed".
Here is a link to a written guide with all of the barrel locations created by Apple on x360a:http://www.xbox360achievements.org/forum/showthread.php?t=12...
For a visual guide, you can use this video from YouTube by AchievementGuides: