Sentence 1 is correct:
He showed his resentment in direct expression of anger.
The article "the" would be used to refer to a specific, previously-mentioned direct expression of anger -- an expression of anger that the reader would have already known about and is being discussed further in this sentence. Because these sample sentences are meant to stand alone without previous context, the "the" ought to be omitted.
Furthermore, there is some ambiguity in the meaning of the sentence. It could mean one of two things:
A. He expressed resentment toward other people's direct expressions of anger. He didn't like it when people showed anger directly and resented that kind of behavior. You could clearly communicate this by saying:
He showed his resentment toward direct expression of anger.
B. He showed his direct expression of anger by resenting the thing that made him angry. You could communicate this by saying:
He showed his resentment through direct expression of anger.
I think it's unclear whether resentment is a direct expression of anger, but I'll leave that open to interpretation.
You can read more about the definite article here:
You can also use "a" as @sovereignsun mentioned, but it's not preferable to no article.