Vic is on the right track, I believe.
Adonijah was a schemer. He had already sought to assume the throne, before David had even died a natural death (1 Ki. 1). He escaped death by a pious appeal to God (v. 49ff). When Solomon pardoned him, however, he clearly took this act of grace as an indication that his younger brother was insecure, that he was not made of the necessary ruthless calibre stuff that kings are made of (that Adonijah himself would certainly have been if he had taken the throne--see 1:21).
In this he was right, but also wrong. Solomon was made of stuff like his father. He was interested in justice first, and his exercise of power was that of a man fully in possession of himself. Adonijah planned to take his father's wife, Abishag, and by this ruthless possession, to challenge Solomon's throne. This subtrefuge Solomon saw through, although Bathsheba was deceived.
It was evident too, that Joab (who supported putting Adonijah on the throne in the first place) was complicit in this move also (2:22). At the least, he incurred treasonous guilt by his previous association with Adonijah, and his fate was tied to Adonijah's continuance in favor. He was already worthy of death, being a murderer thrice over.