The Hittite language is in the Indo-European family, as are Greek, Latin, and Sanskrit, along with such sub-families as Germanic, Celtic, Slavic and Indo-Iranian. The Indo-European sub-family containing Hittite is called Anatolian, and includes several other languages, which in their written form did not survive the Hellenization of Anatolia (modern Turkey) under Alexander the Great and his successors. Some may still have been spoken in rural areas, though none have survived to the present day.
Hittite was written in the cuneiform script during the second millennium BC, as were two other Anatolian languages, Luwian and Palaic. Luwian is also written in a hieroglyphic script of its own. It is plausible that the language of the Trojans was in the Anatolian sub-family, possibly a form of Luwian. Other Anatolian languages, such as Lydian, the language of the Lydian empire under King Croesus, were written in alphabetic scripts during the first millennium BC.
For more details on Hittite see Wikipedia.