In Albert Camus's The Stranger, what is Meursault actually on trial for; his actions or his eccentric character?
In The Stranger, Meursault is initially put on trial for killing the Arab, but the trial becomes more and more about Meursault himself, so the short answer to your question is: Meursault is on trial for both his actions and his eccentric character.
You could make the argument that the court is more intrigued and revolted at Meursault's outlook on life (his character) than the fact that killed another person. This starts out as typical questioning, to find out a criminal's character in order to discover a motive for committing the crime. For example, at the beginning of Part 2, Meursault notes that he does not believe in God and the magistrate finds this "unthinkable." Yet, killing the Arab was not deemed unthinkable; it was just a crime. Meursault's outlook on life is more appalling to the magistrate than the crime itself.
The judge and the Prosecutor spend much more time questioning Meursault about his relationship with his mother than about the murder. The Prosecutor tells the jury that on the day after Meursault's mother's funeral, Meursault was out at the movies with a girl (Marie), later claiming that Meursault is "morally guilty" of his own mother's death. The case becomes all about Meursault's odd behavior. In fact, at the end of part III, in a surprising moment of overt clarity, Meursault's lawyer interjects, asking "Is my client on trial for having buried his mother, or for killing a man?" Shortly thereafter, the Prosecutor responds, saying "I accuse the prisoner of behaving at his mother's funeral in a way that showed he was already a criminal at heart." Judging by this exchange and this last comment, this is a decisive moment and an obvious declaration by the Prosecutor that the jury should focus on the callous way Meursault has behaved, prior to the crime. In other words, the evidence of Meursault's murder of the Arab is an effect of his general criminal behavior.
Meursault is eventually convicted of murder without extenuating circumstances. This is equivalent to first degree murder, willful and premeditated. His lawyer told Meursault to expect a favorable judgment, thinking he might get a few years and a suspended sentence. But since Meursault showed no remorse and since Meursault's philosophy contradicts the God-fearing, moral ideologies of the court, his sentence elicits the maximum punishment: death. Like the trial itself, the conviction is based upon his crime, but the severity of the punishment is based upon Meursault's character.
If this were an either/or question, I would say the trial is more about Meursault's character than the crime. Had he repented and/or claimed self-defense, even though these strategies would have been lies, his punishment would have been far less severe.