What do we learn from "Kel Kano v'Nokem Hashem"?
Rashi: Nachum prophesized about Sancheriv's seed. He was in the days of Menasheh, like Chazal said (refer to 1:1:2:2). From then Hashem took vengeance for Yisrael against their enemies, and He still bears vengeance and anger. He will later avenge His afflicters who destroyed His land and exiled His nation. He still bears His anger at the Bavliyim.
Radak: He has vengeance for His nation - "va'Ykanei Hashem l'Artzo" (Yo'el 2:18), for He sees them in great affliction in their exile. Even though Melech Ashur exiled them based on Hashem's will ? "Hoy Ashur Shevet Api" (Yeshayah 10:5), Ashur increased to do more evil than was proper ? "Ani Katzafti Me'at v'Hemah Azru l'Ra'ah" (Zecharyah 1:15).
Malbim: This is an introduction for the Sefer. Hashem has two Midos. (a) He has Kin'ah for honor of His name. (b) He avenges evil done against Him. For the latter, he does not delay vengeance, lest they do evil constantly and destroy the world. For the former, he does not take vengeance immediately; He stores the anger in His heart and does not carry it out, which is called "Af".
Why does it repeat "Nokem Hashem" two more times?
Rashi citing R. Shimon: There are three expressions of vengeance, corresponding to the three times that Sancheriv exiled Yisrael (Seder Olam 23).
Radak: The matter is repeated to strengthen it.
Who are "Tzarav" and "Oyvav"?
Radak: Here it refers to the kings of Ashur. So Hashem did (He took vengeance) against all who pained Yisrael, like He did to Melech Mitzrayim, Melech Bavel and all who harmed Yisrael. Those who do evil to Yisrael, and Yisrael's enemies, they are Hashem's "Oyvav" and "Tzarav". \h
Malbim: Oyev is one who seeks another's evil, but does not do evil to him. Tzar is one who afflicts in action. You can have a Tzar who is not an Oyev, e.g. he pains for his benefit, and not amidst hatred, and an Oyev who is not a Tzar, if he does not do harm. Regarding Hashem, one who does evil to another person is called Tzarei Hashem, even if he is not an Oyev (he does for for benefit). Oyvav are those who intentionally deny Him and argue with Emunah in Him.
Why does it say "Noter"?
Radak: This is like Shomer. Even though He does not take vengeance against His enemies immediately, He guards the sin until He will take vengeance against them or their children. Kano is an adjective, like "Ki Hashem Kana Shemo" (Shemos 34:14), but it is a different form of the word.
Malbim: He is Nokem (takes vengeance immediately) for Tzarav (those who harm others), and Noter (delays vengeance) for "Oyvav" (deny His Elokus).
