You assume correctly that I and V were the used letters, and as there were no minuscule letters used at the time it would be written as IVLIVS. The pronunciation was not really complicated, there were simply two different pronunciations of those letters depending on the other letters around them just as for example the y (it's a vowel in words like city and a consonant in words like year) or the c (soft in ceiling and hard in cat) are treated in today's English.

The usual way his name was written in an inscription would have looked like C IVLIVS CÆSAR and you would notice that his first name Gaius was also abbreviated with a C as the G was not in use yet; the Æ is not a distinct letter but a ligature of A and E to mainly save space.

Edit: I found a nice looking inscription of the name of Caesar's father (who was also called Gaius Julius Caesar): http://ancientrome.ru/art/artworken/img.htm?id=6433