Nero was not deified by the Senate after his death, which was the only way for an Emperor to truly "become a god". However, during his lifetime, as with other Emperors, he was worshipped as part of the Imperial cult; while inside Rome this was about worshipping his genius (something of a "guardian angel" of the Emperor and his family), outside Rome this did become worship of the emperor as if he were a god. Erker gives an excellent brief summary of this in the Wiley Companion to the Neronian Age.

However, Nero did encourage an association between himself and (in various ways) Apollo and Sol. Both associated with the sun, Apollo was the god of performance while Sol the charioteer of the sun; acting and chariot racing were two of Nero's greatest passions, according to Tacitus, Suetonius and Cassius Dio. Coinage and inscriptions around the Empire reinforced this connection and a colossal statue of Nero was constructed outside the Domus Aurea (Nero's palace), depicting Nero with the iconography of Sol, and Nero was on at least one occasion referred to with the epithet "Nero Apollo" (see Cassius Dio 63.6 and 63.20); see Champlin's Nero, Apollo, and the Poets (Phoenix 57.3/4) for much of this information.

As to whether Nero thought himself a god, however, I would say not. Unlike Caligula (who, according to Dio, openly proclaimed himself a god and asked to be referred to as Jupiter!), Nero trod the Julio-Claudian line on this carefully; the management of his image as detailed above is carefully designed to strongly associate the Emperor with godhood without outright stating it.