Since I've already been talking about Claudius quite a bit this weekend, I might as well field this question too...
Our understanding of Claudius' talents has been shaped by the nature and biases of the literary sources. There are only four that provide (more or less) independent accounts of his reign: Josephus' Jewish Antiquities, Tacitus' Annals, Suetonius' Life of Claudius, and the Roman History of Cassius Dio. Each author had a literary and political agenda, and shaped his portrait of Claudius accordingly.
Josephus, a Jewish protege of the Flavians, was motivated to present Claudius in a generally positive light. Claudius, after all, had been poisoned (or so it was assumed) to make room for Nero; and since the Flavians wanted to blacken the memory of Nero, Claudius was ripe for rehabilitation.
Tacitus, whose great themes were the rise of imperial tyranny and the decadence of the Senate, found ample material in the reign of Claudius, and presented the emperor as an indecisive man driven to tyranny by the machinations of his wives and freedmen.
Suetonius was a biographer, not an historian. In keeping with ancient ideas on the relative functions of those genres, he emphasized the moral qualities and failings of his subject, and accentuated the paradoxical rise of a court fool to the imperial seat.
Cassius Dio, writing in the early third century, presents a basically positive but rather cursory portrait of Claudius, in which the emperor's achievements are his own, and his failings those of this scheming courtiers.
Modern "popular" conceptions of Claudius tend to be dominated by Robert Graves' I, Claudius and Claudius the God and the BBC series based on them. Graves drew heavily on Suetonius, who provides a fascinating wealth of detail about the emperor's odd personal mannerisms. The idea that Claudius was a secret genius really grows out of Graves' novels, which show a highly intelligent but awkward man trying to survive in a vicious imperial household.
So where does the truth lie? It is clear that Claudius was a competent administrator - his recorded measures are reasonable, and occasionally even far-sighted. It is equally clear that he was a bumbling politician, prone to being used and conspired against.
One of our very few unvarnished views of Claudius comes from the so-called Lyon Tablet, a bronze panel that records verbatim a speech the emperor made in the Senate to advocate citizenship. This is our one real glimpse into how Claudius always spoke - and it is fascinating. The emperor rambles on about historical minutiae, and has to be reminded repeatedly (by senators shouting from the benches) to get to the point. We see, in other words, an obviously intelligent and learned man, but also one awkward in manner and speech. The Lyon Tablet seems to more or less confirm Suetonius' comments on Claudius' behavior as emperor. Claudius was definitely smart - but we should not imagine him as a secret mastermind.