A number of the above answers are correct, but I sense from @Aethelflaed‘s “bartender put something more into the glass” that some deconstruction might help.
A chaser is an additional drink (it chases the first drink).
Hemlock is a well-known poison and no one uses it to make drinks.
To say, give me x beverage and a hemlock chaser is best understood as @FutureMemory described above.
One could make a similar joke (albeit not that funny) in a situation with an older man and a younger woman going into a bar. The man orders her a vodka and the bartender asks if she would like a Milk chaser with that (poking at her age).
There are any number of liquids (including those that one would never drink) that could be added to the “Give me a [fill in the blank] chaser” statement to poke fun at yourself or others in comedic fashion.
The original purpose of a “chaser” is to provide a mild beverage immediately following the consumption of a stronger, alcoholic beverage. This could be to cleanse the palate, reduce a burning sensation, etc.