The existence of a "First Catilinarian Conspiracy" is at best considered doubtful and more usually considered to be a total fabrication. As for the plot of 63, Caesar probably had little or nothing to do with it. Caesar had been a political ally of Catiline, overseeing his acquittal as the presiding judge in his trial in 64. But so had Cicero--during Catiline's trial de repetundis in 65 Cicero had considered defending him, but eventually declined. During Catiline's consular candidacy either Caesar or Crassus probably supported him, based mainly on the oblique references in Cicero which Asconius took as meaning either Caesar or Crassus. Crassus, Asconius asserted (on the basis of Cicero), had been involved in the plot of 65, but there is very little reason to believe that this plot ever existed and in any case Crassus, though Caesar's important ally until Pompey's return from the east, was not Caesar. It's certainly possible that Caesar was in on the Catilinarian Conspiracy, and without some kind of smoking gun evidence we cannot definitively prove that he was not. But, notably, Caesar played a crucial role as an informer against Catiline, despite the attempt by Caesar's opponents to bring him to court for dealings with the Catilinarians--Cicero's testimony that Caesar had aided him by reporting on the Catilinarians fully discredited Vettius' accusation against him. It's possible that Vettius really did have good information, but even if Caesar had been in collusion with the Catilinarians at some point by November of 63 he had certainly turned towards Cicero at least in some measure. Moreover, Caesar (and, to a lesser degree, Crassus) turned out to be pretty instrumental in the senatorial response to the Catilinarians. Notably, both Caesar and Crassus were placed in charge of imprisoning Statilius and Gabinius, respectively--remember that Rome had no provision for public incarceration while awaiting trial, and it was up to the accused (with the oversight of a private citizen or magistrate) to appear in court or be automatically condemned. Though Caesar was praetor elect at the time both he and Crassus were private citizens, and trusted by Cicero enough to keep watch over two known conspirators. Much of the suspicion regarding Caesar seems to derive from his opposition to Cicero's suggestion to execute the Catilinarians without trial. If Sallust's version of the speech bears any resemblance whatsoever to the reasoning Caesar actually gave then Caesar's logic was, by any reasonable standards, impeccable. Suetonius gives a paraphrase of the speech, noting that Caesar pointed out the resentment that the plebs would have against Cicero and the senate if the leges Porciae and the lex Sempronia were violated. And indeed, Cicero faced opposition from the tribunes of the plebs as a result, when he was not allowed to deliver his consular parting address due to tribunate vetos. Caesar's opposition to Cicero's suggestion and his own proposal for the punishment of the Catilinarians should not be taken as evidence that Caesar was involved in the conspiracy, but rather that in a moment of senatorial panic Caesar managed to keep a cool head and refused to violate one of the most fundamental principles of the Republican state, the right of the citizen to trial. And Caesar was hardly the only one, since Caesar's counter-proposal swayed many senators to his side, including Cicero's brother Quintus (fellow praetor-elect and later a Caesarian, though he would join his brother in Pompey's camp during the civil war), until Cato demanded that the Catilinarians be executed immediately, trial or not.

While we have little way of knowing for sure how involved Caesar was in the Catilinarian Conspiracy, there's really very little evidence that he was. He had been a supporter of Catiline's consular bid, but he seems to have gone out of his way to distance himself from Catiline following his failure at the elections. It's possible to interpret Caesar's outrage when Vettius accused him as a tacit admission of guilt, but it's also very much in keeping with Caesar's personality, as his obsession with dignitas seems to have caused him to take any personal accusation or slander, no matter how minor (such as the jeers of the crowd during a theatrical exhibition during his consulship, when the audience also famously applauded the line nostra miseria tu es magnus as a dig at Pompey)