I know that Christianity would go on to become the faith of the Roman empire in its later years, but did the emperor of Rome at the time (or any of his administrators) know about the growing following of Jesus?
No, not so far as we have any evidence.
The life of Jesus occurs between 4 (perhaps up to 6) BC, and 33 AD (some place his date in 30AD, I think 33 is more likely). In any case, that put it entirely into the reigns of Augustus (ruling through to 14AD) and Tiberius (ruling through to 37AD).
There are no documentary or inscriptional evidences suggesting that either of these emperors had knowledge of Jesus or his first followers. They may well have, but that is not substantiated by our data.
Technically Pilate counts as an leader of the Roman administration.
There's every reason to think that Roman procurators and prefects in the area of Syria would have knowledge of early Christians, but again this is not evidenced in Roman historical writings. Pliny is really the best and earliest evidence of a "Roman leader" dealing ith Christians, during his time as legatus Augusti in Bithynia and Pontus from 110 onwards.