If you don't have DLC pack two, you will need to grind this achievement out by replaying the sewers section of I Now Know Why You Cry. Simply blow up the two dogs by shooting them, or using a grenade, then kill yourself and repeat until you have all 13 kills.
Alternatively, if you have DLC pack two, start up The Places We're Meant to Die. At the very beginning of the level, you'll have a dog running toward you from the hospital parking lot, and you'll quickly come into another one behind the last car in the parking lot. Kill both dogs and run into the fire at the end of the parking lot to kill yourself and repeat.