Use "after which" to connect two parts of a sentence across a COMMA ( , ) . Use "after that" to connect two DIFFERENT sentences across a PERIOD ( . ).
I'll buy dinner, after which we can go to a movie.
I'll buy dinner. After that, we can go to a movie.
This is related to a similar rule: you should almost never use "that" after a comma.
This is the sandwich, which Sam ate, and this is the cake, which Bob ate.
This is the cake that Sam ate. It isn't the cake that Bob ate.
For more on this rule, search for "restrictive and nonrestrictive clauses".