Lets say that at my opponent's end step I cast Bile Blight on his Pack Rat that he already has 3 of total on the field, then he activates the ability of pack rat in response, can I respond to the new pack rat resolving with Golgari charm and kill all 4 rats?
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Yes, though it happens in a complicated way.
Golgari Charm's first mode creates a continuous effect that modifies characteristics (power and toughness). It only affects the creatures on the 'field when it resolves.
(Same goes with Bile Blight. Any Pack Rats entering the field after it resolves won't be affected by it.)
That means the following will happen:
It's a little cleaner if you wait before casting GC.
No matter the order, creating a 5th PR before BB resolves would save them all. But which of the approach you used will affect how strong they are.
Here what will happen.
Now, if you waited for the pack rat ability to resolve first, then the 4th would end up getting 1 more -1/-1 which although useless in this case, may be relevant if your opponent has more spells. So it is worth doing like this (and it seems this is what you meant in your question)