The volatile mob in the courtyard, mostly idlers that gathered whenever the Senate met, initially approved of Caesar's assassination. They dragged his body to the forum, halting at the forum's western end in front of the new rostrum, the speakers' platform, which Caesar himself had built and decorated with the bronze rams of the ships of the Pompey's defeated fleet. Mounting the dais, Marc Antony delivered the stirring speech that turned the mob around. Now Caesar was the fallen hero, deserving of a hero's cremation. A makeshift altar was raised on the spot and furniture from surrounding temples and basilicas made the pyre.
Within a few weeks, a more permanent commemorative altar was in place and a twenty-foot high column of rare Numidian marble was raised with the inscription "Father of the Country". Altar and column were soon replaced with an even more grandiose structure approved by the Second Triumvirate (Octavian, Lepidus, and Antony) but actually built only by Octavian. Dedication of the new temple was delayed until Octavian was installed as sole ruler in 29 BC after defeating Antony. (Lepidus had, quite sensibly, retired voluntarily and avoided the fray.)
When Octavian, soon to be called Caesar Augustus, built the temple dedicated to his uncle, Julius Caesar, he changed completely and forever the character of the Roman forum. The Temple of Julius Caesar was, obviously, dedicated to Julius, but it really was an expression of the supreme power of Octavian and of his expectation that his family, the Gens Julia, would permanently rule Rome.
Until then, the broad open space in the center of the forum had stretched from the civil government buildings on the western face of the Capitoline hill to the sacred complex of the Vestals and the Regia, the residence of their protector and sponsor, the Pontifex Maximus, the high priest of the Roman religion. Octavian's "Temple of the Divine Julius", the first of the Roman temples to deified emperors (and, later, spouses) to be built in the Forum, blocked out the Vestals and the Regia and imposed on the western end of the Forum a new "Julian" (that is, Gens Julia) religious flavor.
Worship of the high gods -- Jupiter, Juno, and Minerva, and their family and supporters -- had already become almost purely ceremonial in the late Republican period. Now religion was to be revived based on the worship of dead and deified Emperors, and this marked a movement of the still current ancestor worship from the privacy of the home literally into the public forum. There is no doubt that this was, at least partly, self-serving on the part of Octavian and the Gens Julia. Later Emperors, of course, quickly decided that it was better to be gods while still living -- though their temples were still usually built after they died, they wanted other benefits of apotheosis sooner rather than later.
Temples in the Forum were built on high platforms to avoid damage from the periodic floods of the area. The platform of the Deified Julius Temple was 3.5 meters high, 26 meters wide, and 30 long. A second platform, another two and a third meters higher and about 17 meters square, was the base for the temple itself. The Temple and its pillared porch covered the upper platform and, conforming to rules of temple architecture, it would have been over 20 meters high. Some modern representations show six equally spaced pillars, but the physical remains show that the two middle pillars were really wider spaced than the others. A colossal statue of Julius Caesar with a star or comet on its head, perhaps a reference to the comet that reportedly appeared shortly after Caesar's death, stood inside and at the back of the cella, the single room of the temple. This could be seen from outside between the two wider-spaced central pillars.
Two stairways up to the first level widely flank a semi-circular niche built to accommodate the site of the original altar where Julius had been cremated. Above and around the niche was Octavian's expansion of Julius Caesar's rostrum, from which Marc Antony had eulogized Julius and swayed the mob. To remove any doubt the people might have concerning what this new temple was really about, Octavian replaced the rams of Pompey's ships that had decorated Julius Caesar's rostrum with those Octavian and his admiral, Agrippa, had captured in the decisive defeat of Antony at the naval battle of Actium. A large statue of Julius Caesar stood in front of the temple on a separate pedestal built on the level of the floor of the Forum.
What's left today includes part of the wall of the rounded niche. Inside the niche is the weathered cement core of the crematory altar, and the fact that it is slightly off center leads experts to believe that it actually marks the spot of the cremation -- an altar that was merely commemorative more likely would be centered in the niche. A more recent wall has been built across the front of the niche allowing access only on the right hand side. The niche is covered with an ugly modern tin roof to prevent further weather damage. Behind the niche is the massive cement core of the platform, from which you can get an elevated view of the center of the Forum if you can scramble up before the officious "Forum patrol" chases you off. There is still some brickwork on the core, but all the stone long ago disappeared into newer Roman monuments. The base of Julius Caesar's statue is one of the stone piles in front of the platform and altar, but I haven't figured out which one -- some drawings show it as the big one centered in front of the niche..
Description from the Platner and Ashby Topographical Dictionary: http://www.ukans.edu/history/index/europe/ancient_rome/E/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/.Texts/PLATOP*/Aedes_Divi_Juli.html
Descriptive text (in German) with a good site map and plan: http://www.roma-antiqua.de/rundgang/pages/caesartempel.html
Pretty much what you see today -- a different uglier roof is over the central niche: http://www.geocities.com/~stilicho/virtual/temple_caesar.html
From the Seindal Archive: http://www.seindal.dk/rene/sights/sight/171.htm
P.S.: 1. Julius Caesar's crematory altar is still usually decorated with flowers, which, by some accounts, are placed there daily by monarchists and neo-fascists.
P.S.: 2. Antony's speech was not the one everybody knows from Shakespeare. Those famous and stirring lines -- "Friends, Romans, countrymen, etc." -- are Will's alone. Nobody took down the real speech, but it must have been a dilly. Antony, of course, was not just eulogizing Caesar but was rehabilitating him and doing so specifically in order to make sure that Caesar's last will and testament, to be read a few days later, would be carried out. Antony had been the very successful and much liked governor of Rome while Caesar was pursuing Pompey and, later, while Caesar was dallying with Cleopatra. He had every reason to believe that he would be named in the will as one of Caesar's chief heirs -- the other possible heir, Brutus, was in flight for participating in the assassination. Imagine Antony's surprise when Octavian was left almost everything. Some of Antony's partisans called for a recount saying that Octavian had somehow "fixed" the will in cahoots with the Vestals, the custodians of official wills. That's unlikely, however: Octavian was still a very young man, who wasn't even living in Rome, and he wouldn't have had much clout at the Domus Vestalis.