Why does it say that his words were with Yo'av and Evyasar?
Radak: He took counsel with them. They wanted that he will rule, and persuaded David's other sons and Anshei Yehudah to support him - "va'Ya'azru Acharei Adoniyahu."
Malbim: Because Yo'av, the head general, supported him, the army would go after him. Because Evyasar, the [former] Kohen Gadol supported him, the common people would support him. Therefore, Adoniyahu was not afraid to declare himself king.
Why did Yo'av support Adoniyahu?
Why did Evyasar support Adoniyahu?
Rashi: He was deposed from Kehunah Gedolah when David fled Avshalom, and Evyasar was not answered through the Urim v'Tumim (Shmuel II, 15:24). He descended from Eli; he knew that he will not serve in the days of Shlomo, for Eli was told "I will establish a faithful Kohen for Me? he will go in front of My anointed" (Shmuel I, 2:35). He wanted that Adoniyahu will rule through him.
Mahari Kara (2:27): Hashem put this in his heart, in order to fulfill the curse against Beis Eli. 1 He should have known that David will not consent to divert kingship from Shlomo, for the Navi said that Shlomo will succeed David!
Malbim citing Mahari: He saw that David shows more honor to Tzadok than to Evyasar.
Why would the curse apply when Shlomo is king, but not when Adoniyahu is king? Perhaps he understood that Shlomo is considered "My anointed", but Adoniyahu is not. Or, Adoniyahu will reinstate Evyasar amidst appreciation for helping him to reign. (PF)