The assassination of Caligula has always reminded me of that "Who Shot Mr. Burns?" Simpsons episode, in that just about everyone in Rome seemed to have a reason to kill the emperor. Caligula had terrified the Senate, antagonized the Praetorian Guard, and convinced even his natural supporters (i.e. his freedmen) that he was too unstable to tolerate. According to Josephus, who provides the fullest account of the assassination, three separate groups of conspirators cooperated to allow Cassius Chaerea (a brave Praetorian officer with the unfortunate handicap of a squeaky voice) and a small band of collaborators to cut Caligula off from his loyal German guards and stab him. Then, after some confusion, Caligula's long-suffering and generally-ignored uncle Claudius was, for want of anyone better, proclaimed emperor. As Suetonius has it:

"In great terror at the news of the murder, [Claudius] scurried to a nearby balcony and hid among the curtains which hung before the door. As he cowered there, a common soldier, who was prowling about at random, saw his feet, intending to ask who he was, pulled him out and recognized him; and when Claudius fell at his feet in terror, he hailed him as emperor." (Claud. 10)

By the traditional account, in short, Claudius was a hapless stooge, who fell bass-ackwards into the imperial throne. As a ruler, however, he proved surprisingly competent, if rather eccentric. So did he have a hand in the assassination? Was he like Lucius Junius Brutus, who was said to have survived the caprices of Rome's tyrannical last king by playing dumb until the time was right?

It is not impossible that Claudius played a part in Caligula's murder; but if he did, the secret was very well-kept.

In the months and weeks leading up to the assassination, Claudius was recognized as a potential successor; Josephus tells us that the imperial freedman Callistus had begun to cultivate his favor, correctly assuming that he would come to power once Caligula was gone (AJ 19.1.10). But if Claudius was aware of the conspiracy - and that, I think, is about the highest degree of involvement we can plausibly assign him - Josephus is silent. It is interesting, if perhaps unsurprising, that Caligula suspected Claudius of plotting against him (Suet., Claud. 9) - though the fact that even Caligula could find no sufficient reason to have him killed is probably significant. Even more interesting is the fact that Claudius left the theater just before Caligula was assassinated - possibly a coincidence, but just as possibly evidence of foreknowledge.

Other circumstances, however, suggest that Claudius was at most passively aware of the conspiracy. Shortly before his assassination, Caligula threatened to execute a certain Capito, who attempted to save his life by naming every person close to the emperor who might reasonably be accused of plotting against him (Cass. Dio 59.25.6). Claudius' name was not on that list. There is no evidence, moreover, that Claudius was popularly thought to have been involved. Shortly after his death, Seneca wrote the infamous Apocolocyntosis, a savage (and very odd) satire about Claudius. He accuses him of many murders - but not of Caligula's (in fact, Caligula shows up at the end of the satire and claims Claudius as his slave).

We might imagine Claudius being aware that a conspiracy was underfoot; we might even conjecture that some of the ringleaders had contacted him. But I think it very unlikely that he was pulling the strings.

EDIT: added material to the beginning of the answer