Make your enemy as normal, but don't set the deathstate yet. Go to the Things Editor and look for the thing you want the enemy to drop. Take note of it's thing number. Now go to the state editor and make four states (I will refer to these as A, B, C, and D). Set them up like this:
State A:
Sprite: 82
Frame: 0
Tics: 1
Function to call: A_Scream
Var1: 0
Var2: 0
Translucency: 0
Next State: Whatever you have for B
State B:
Sprite: 82
Frame: 1
Tics: 1
Function to call: None
Var1: 0
Var2: 0
Translucency: 0
Next State: Whatever you have for C
State C:
Sprite: 82
Frame: 2
Tics: 1
Function to call: None
Var1: 0
Var2: 0
Translucency: 0
Next State: Whatever you have for D
State D:
Sprite: 82
Frame: 3
Tics: 1
Function to call: A_SpawnObjectRelative
Var1: 0
Var2: Whatever the object number is for your thing.
Translucency: 0
Next State: 314
Now that you have that done, make your enemy's deathstate State A.
You're done!