It's unlikely that Pilate would have reported the execution to Tiberius and Tacitus, our earliest source for Imperial activity against Christianity, does not say that Tiberius knew anything about it - simply that it was during his reign (Tibero imperitante). Based on the same data, Nero could have known about Jesus, but whether he did and how much he would have known is not known to us. All Tacitus tells us is that Nero blamed the Christians for the fire in Rome (ergo abolendo rumori Nero subdidit reos et quaesitissimis poenis adfecit, quos per flagitia invisos vulgus Chrestianos appellabat - Annals 15.44). Caligula seemed to have thought of Christians as a subset of Jews. This persisted until 96, when Nerva re-vamped the tax laws in Judea and separated Jews from Christians therein (cf. Goodman, "Nerva, the fiscus Iudaicus, and Jewish Identity," Journal of Religious Studies 79 (1989):40-44).

Trajan, Nerva's adopted son and successor, may have known something about Jesus - but, again, in his correspondence with Pliny (Ep. X.97), the governor of Bithynia, he mentions Christians, but not Jesus. He tells Pliny not to hunt Christians, but if one is brought to him, to punish him - unless the Christian recants. Tertullian later would mock this way of dealing with 'criminals' (Apol. 2). When I searched Brepolis' Library of Latin texts, Tertullian was the earliest Latin usage of "Iesus" I could find - that is to say that in the first couple hundred years of Christianity, no Roman emperor mentions Jesus by name in his extant writings.