How can I tell if I am drunk and how long will this last?

After consuming alcoholic beverages your screen becomes blurry, it is only while the screen is blurry that kills count towards this Achievement (~5-10 seconds). The best place to get this is when you first arrive in Shantytown, there is a bar at the bottom left. Elizabeth will make a comment that you could either go to the bar or go on with the quest. As soon as you enter the bar you will see the autosave icon display. In here there is plenty of alcohol you can consume and killing any of the NPC's will count towards the Achievement. If you run out of NPC's or out of Alcohol you can reload and repeat the process until the achievement pops.