The Roman Republic during the tetrarchy between Mark Antony, Octavian, and Lepidus was essentially three separate states with a very loose alliance binding them. The Senate was unable to capitalize on the assassination of Julius Caesar, and with most of the legions supporting the Triumvirate, it fell into powerlessness. It is important to note that Lepidus gave up most of his legions shortly after the formation of the alliance in order to help defeat senatorial loyalist under Brutus. This would leave him the weakest of the three.

To your second question, Mark Antony claimed that Julius Caesar wanted him to have power due to the fact he was one of Caeser's must trusted allies (example: Antony was give control of Italy for a short time during Julius Caeser's reign). Octavian staked his claim on Caeser's last will. Lepidus was also a close ally of Caeser's, but never really attempted to take power due to lack of soldier/ambition.