The religion that Julian was trying to promote as a rival to Christianity was more than the rituals of the traditional Greek and Roman worship of the gods, it was highly influenced by the Neoplatonic philosophies proposed by Plotinus, Porphyry and Iamblichus. Neoplatonism is a modern name for the schools of thought based on Plotinus' interpretation of Plato's writings and his attempt to synthesise them with other, later, philosophical schools.
Very basically, the idea was that the Divine was so caught up contemplating its creation that some parts got trapped inside the creation and now have to find their way back through the study of philosophy. How exactly this was to be done differed from writer to writer. Iamblichus invoked the aid of the gods, who were supposed to be parts of the Divine who were less trapped than humans, using a technique called theurgy.
Of course, there was a lot more to it than that, and the Neoplatonists have attracted more than their fair share of those with very individual views of history, philosophy, and religion.
Eunapius' biographies include not only Plotinus, Porphyry and Iamblichus, but also Julian's teachers and stories of his studies. The Internet Enclyopedia of Philosophy has a general article on Neoplatonism and one on Plotinus (neither of which is for the faint-hearted). The Stanford Encylopedia of Philosophy has slightly easier articles on Plotinus and Porphyry. For an account of Iamblichus there is an article in Wikipedia (even more so than usual, caveat lector). Plotinus' Enneads as edited by Porphyry are online, as are many of Porphyry's own writings. Iamblichus is represented by his On the Mysteries.
Two of Julian's philosophical/religious orations (one on the Sun, and one on the Mother of the Gods) are online in two different translations, a late eighteenth century one and a nineteenth century one (separate pages for the Sun and for the Mother of the Gods).