Hooray! I cover this in my term paper!

First off, this thread is probably exactly the thing you're looking for. Second, when you say Holy Roman Empire, are you referring to Charlemagne and the Carolingians, or the Holy Roman Empire of Otto I? They weren’t happy about Charlemagne, and even fought a war with the Carolingians over Venice. When Otto I was crowned, relations between Byzantium and HRE immediately grew more intensive. Ultimately, the Byzantines came around because: the Byzantines were already used to the idea of a Western Emperor after Charlemagne and Louis the Pious, and they were looking for an ally against the Fatimids in Egypt. The Emperorship was recognized and sealed with a marriage between Otto’s heir, Otto II and niece of the Byzantine Emperor.

Funny enough, it's unclear that Otto I ultimately benefited from being crowned Emperor, and it may have even been a liability. It’s likely that Emperorship was thrust on Otto by the pope, who needed a strong ally in Italy. There's also a similar view regarding the circumstances of Charlemagne being crowned.

source: Reuter, Timothy. "The Kingdom of the Franks and Saxons: Otto I and Otto II." In Germany in the Early Middle Ages, C. 800-1056. London: Addison Wesley Longman Limited, 1991. 172-174.

edit: grammar, wording

The Byzantines were called Romans. In Greek it was Basileia ton Rhomaion (basileus meant something akin to king or ruler, so this meant something like Empire or Kingdom of the Romans). I think the word Romanoi is still used in Greek even today. There was no reason to stop calling themselves Roman because in their minds they were still the same empire. Constantine had made Constantinople the overall capital well before the West fell, and being Roman was more a particular way of life which they saw as civilised than a nationality. Furthermore, when the Ottomans conquered the Byzantine Empire they called the Balkans Rumeli, and the title Sultan-i Rum was sometimes used. When they referred to their Orthodox Christian subjects they called them 'millet-i Rum' or the Roman Nation (nation meant more a group of people than the more modern State definition). 'Byzantine' was a word made up for them later by a German in 1557.

As to how they viewed there being a 'Roman' emperor in the West, remember that whilst the Byzantine empire has just kept going, in the meantime the West devolved into Germanic kingdoms. Before the 12th century renaissance Western Europeans were known for their tough fighting skills and not much else. They were employed as the Byzantine Emperor's mercenary bodyguard called the Varangian Guard. Imagine you're part of what is still known to you as the Roman empire, which while it lost its Western half is still going after another several centuries, and all of a sudden you get some people from that half which was overrun by 'barbarians' calling themselves Western Roman Emperor. You would be somewhat incredulous. They had to put up with it, but I think they rather looked down on it. Even in the eleventh century they still didn't think much of Western Europeans.

I found this to help get you started:

• How did the Byzantines view the Holy Roman Empire and vice versa?

As always, further discussion and new answers are welcome and appreciated.