Ghost, Dark and Bug.
A recommended moveset for Azelf is:
Thunderbolt
Flamethrower
Psychic
Shadow Ball
That deals with all of it's weakneses.
Chat with our AI personalities
Ghost & Flying. I believe that's it! Not dark or bug because although they're super-effective against Psychic, they are not-effective against Fighting. And not Psychic, because though it's super-effective against Fighting, it's not-effective against Psychic.
Azelf is a legendary Pokemon.
electric and ice are effective against flying
Water and Grass attacks are super effective against Onix.
If you knock out Azelf in Pokemon Diamond, Azelf will be gone forever (unless you have saved before battling Azelf and shut off the game and turn it back on to try again).