In one sense, Falkland displays a belief in right and wrong being relative, meaning that what can be judged as right and wrong depends on a person's perspective. One way in which he portrays this is through his evident beliefs in the darker honor code Renaissance Italians carried out, such as the idea that duels and even murder are acceptable ways to redeem a person's lost honor. In Chapter II, we learn that Falkland was fond of the "heroic poets of Italy" and learned from the poets "love of chivalry and romance." Enotes editor Scott Locklear informs us that the Renaissance Italian poets he is referring to are those such as Boiardo, Ariosto, and Tasso, poets who promoted duels, assassinations, and even war above classic Arthurian chivalry (eNotes, "Extended Summary"). In Italy, when Falkland is accused of seducing Lady Lucretia, characteristic of the heroes in his favorite poets' works, Falkland warns Count Malvese to avoid making hasty accusations, "the consequences of which may be inexpiable but with blood," which is to say that one may not be able to atone for making hasty accusations, or dishonoring another person, except by being killed, as in a duel (Ch. II). Hence, one thing that can be said is Falkland believes in a different honor code from others in believing one can redeem one's honor through killing another person, as in a duel; therefore, Falkland does not necessarily have the same beliefs in right and wrong that others do.
Later, we do see Falkland refer to his murder of Tyrrel as a "crime" and be tormented with what looks like feelings of guilt. However, we also hear Falkland declare to Caleb that "honour, justice, virtue, are all the juggle of knaves!" (Vol. II, Ch. II). He even further declares, "Justice! ... I do not know what is justice. My case is no within the reach of common remedies; perhaps of none" (Vol. II, Ch. II). If we remember his reasons for killing Tyrrel, which were fist Tyrrel's cruelty towards Emily and then Tyrrel's public humiliation of Falkland, we can judge from these declarations that Falkland doesn't truly believe that things like "honour, justice, [and] virtue" really exist. For instance, it was certainly not honorable for Tyrrel to treat Emily with such cruelty, and it was certainly unjust for Tyrrel to be indirectly responsible for her death and receive nothing more in the way of punishment but a reprimand from the county Assembly of noblemen and a retraction of his title and privileges. It was also certainly not honorable for Tyrrel to physically abuse Falkland for having testified against Tyrrel's behavior, thereby dishonoring Falkland. Hence, we can certainly see from Falkland's perspective that things like justice and honor are really mere illusions and also understand exactly why Falkland felt compelled and even justified in enacting the Renaissance Italian honor code by murdering one who had offended not just his own honor but helpless Emily's honor as well.
However, he does call himself the "blackest of villains" and behave as if he is being driven to insanity by feelings of guilt. So, we can't completely argue that Falkland felt his actions to be "right." At the same time, Falkland also seems to blame his actions on fate, as he later declares, "My virtue, my honesty, my everlasting peace of mind, were cheap sacrifices to be made at the shrine of this divinity," meaning God (Vol. II, Ch. VI). In other words, it was God's intention to take everything that was good about Falkland and turn it into something cheap and useless.
Hence, though Falkland felt a sense of guilt about his crime, he viewed right and wrong as relative. He did not truly believe in the existence of things like justice, virtue, and honor, because such things are not truly preserved. Instead, he believed in enacting the Renaissance Italian honor code to carry out justice and preserve honor, a code that endorses duels and assassinations, in the event that nothing else could be done to ensure the preservation of honor and justice.