The Byzantines always wanted a restoration of the territory of the Roman Empire. On many occasions, emperors sent expeditions to reconquer Italy (Justinian), Spain (Justinian), North Africa (Justinian), Southern Italy (Kommenian dynasty), Sicily (Macedonian dynasty), interior of Anatolia (Kommenian dynasty and pretty much everyone post 1071), and a bunch of other places. Egypt was the target of a Western and Byzantine allied crusade in the 12th century. In fact, the initial crusades were encouraged by the Byzantine Emperor as a way of getting the Latins to help expand Byzantine territory, in particular the area around Antioch. Didn’t quite work as planned. For revanchism, the great dream of every emperor was to be a Justinian, and reconquer the lands that were “rightfully” theirs.

The Byzantine state was the same continued political entity as the Roman state. It is better to think of the Byzantine Empire as the last era(s) of the original Roman Empire. Sure, they stopped using Latin as the official language by the 7th century, but they always saw themselves as the true Roman empire, all the way through 1453; many Roman traditions remained in place for well over its thousand plus year history.

Lastly, the Byzantines did not see the Holy Roman Emperor as the "Roman" emperor. The first one was Charlemagne, and the Pope crowned him, which became (another) source of anger between east and west. Prior to Charlemagne, the West, including the papacy, acknowledged the Byzantine emperor to be the one true Roman emperor.

I highly recommend reading a Short History of Byzantium by John Julius Norwich for an overview of the many reconquests of the Byzantines, among other things.

EDIT: Thanks /u/Guckfuchs for your clarification on the "Roman" aspect of the imperial title.

Great question, paging one of our resident Byzantinians /u/B-Hosk

You may find this post somewhat relevant to your question.

For the first question, about whether the Byzantines saw themselves as Romans, I will refer you to a previous answer in which I partially covered it.

Non-Byzantine (e.g. Jewish and Arab) sources also mention Byzantium as "Rome".

The guys belonging to the IS who now threaten Rome are oblivious of this fact, namely that when their Arab source mention Rome, they are referring to modern Instanbul, not the one in Italy which was not relevant at all at the time. Ironically, Turkey's government seems rather ambiguous towards them, but that's another story.

Back to your question. After the fall of the original Rome, they felt to be the legitimate heirs of the Empire, and considered Byzantium as the second Rome: like the first one, they had their own church, their own bureaucratic apparatus, etc.

Incidentally, after the fall of Byzantium, Moscow would consider itself the Third Rome (again, its own church, etc.). This is why, for example, the Russian emperor was called "Czar", from "Cesar".