This achievement's description is quite confusing, as one would think you would have to be the one who is hungover while getting a kill.
That's not the case. All you must do is kill an enemy player who is effected by your Hangover Burst. Be sure to get your Burst meter to Level 3. A Level 3 Hangover will cause enemies to spawn with little stamina and low health for about 10-15 seconds.
It is suggested that this achievement be boosted with a full 8-player lobby, because it is likely that an enemy will hold back until their Hangover wears off in a legitimate match. A great map for this achievement would be "Branco HQ." Activate your Level 3 Hangover burst and kill your partner. When they spawn, communicate quickly in order to locate and kill them while the Hangover is still in effect on them.