Cæsar’s name is properly and rightly rendered in the Classical Latin of his day as GAIVS IVLIVS CÆSAR (i.e. the late form j is not used, both because of its late appearance in the 16th century and because it is not pronounced differently from the i and y in Latin, while the differentiation between the v and the u is similarly of late appearance--in the 14th century--and would have been unrecognizable to a Latin speaker of Cæsar’s day). The antique prænomen form CAIVS is also attested and is effectively interchangeable, since in employs the ancient pronunciation of the letter c as g, as in English ‘goose’ as an adaptation of the Greek γ.
In regards to the proper Classical Latin pronunciation of CÆSAR, the c prior to the diphthong æ is properly pronounced like the hard English k, as the c in ‘cat’, or the k in ‘keep’. The diphthong æ is properly pronounced separately, i.e. with the a pronounced like the in the English ‘marble’ and the e of the diphthong is pronounced like the English stressed è, like the e in ‘bed’. The s, since it is preceded by a stressed e is pronounced like the z in ‘zebra’ not, as conventionally, like the s it ‘sit’, and the second a is again pronounced like the a in ‘marble’, and the r like r in r in the American English ‘roar’, thus the proper phonetic pronunciation is something akin to ‘kaè-zar’. Similarly, in the koinē Greek lingua franca of Cæsar’s day, his name was written as Καίσαρ, which properly transliterated as kaisar, with the Κ hard as as in ‘stick’, the diphthong αί as in ‘aisle’, the σ as in ‘sister’, the α as in ‘father’, and the ρ trilled like the Spanish r as in ‘rapido’. Note that this is very much a vexata quæstio, i.e., a disputed question. The diphthong æ is generally rendered more simply as in the English ‘eye’, but the above is, in this case, a more proper and accurate phonetic rendering, and is supported by comparison with the German kaiser.