Let's begin with the deification itself, here's what Suetonius tells us:

Instead of a funeral panegyric, the consul Antony ordered a herald to proclaim to the people the decree of the senate, in which they had bestowed upon him all honours, divine and human; with the oath by which they had engaged themselves for the defence of his person; and to these he added only a few words of his own [---]. (Suet. Jul. 84)

Basically, there was no real standard for being deified in the Roman state at this point in time, so while we use the word worship today, it isn't exactly correct in describing what the Romans did. As you can see from the quote, it really means that he was paid the highest respect in form on divine and human honours. That would mean that he would be paid these respects both in private and by the state - as is shown by the erection of a temple to Divus Julius in the city of Rome itself after Caesar's death.

Essentially, broken down to layman's terms, this nomination by the senate meant that Caesar was declared a god was to be issued a temple and a priesthood to go with it (Mark Anthony actually became the first priest of the cult of Caesar). With this nomination follows a standard procedure for worship, with state sacrifices taken care of by the Roman state, as well as other celebration on important Roman holidays. In the case of Caesar's cult, the ides of March obviously would've been an important day.

What this meant for private worship is described in the next chapter of Suetonius' work:

They afterwards erected in the Forum a column of Numidian marble, formed of one stone nearly twenty feet high, and inscribed upon it these words, TO THE FATHER OF HIS COUNTRY. At this column they continued for a long time to offer sacrifices, make vows, and decide controversies, in which they swore by Caesar. (Suet. Jul. 85)

The private cult in itself is something that is very hard for us to grasp completely how it worked. When it comes to state practices, we can rely on materials that withstand the test of time - temples and altars are often made in marble. However, while speculative, the private cult likely included miniature altars and models of Caesar made of things like wood or clay. The former would obviously have desolved over time, while the latter likely has been shattered to pieces or at least worn to a state where we can't see who it's supposed to be - if indeed they were well made at all. There are actually indicators towards items like these existing in the form of silver altars and temples that have been preserved, and might have belonged to the richer parts of society.

If you're further interested in the kind of sacrifices that would be made, I can recommend Simon Price's article "Between Man and God: Sacrifice in the Roman Imperial Cult" (1908) in Journal of Roman Studies, vol. 70, pp. 28-43, available through Jstor.