The Byzantine Empire like the Western Roman Empire didn't fall solely to external struggle but internal struggle as well. The Byzantine Empire instituted a theme system that lasted about three centuries before it was undermined in the eleventh century. The Theme system (in case you didn't know) separated the empire into themes and each theme would be managed by a general that would be appointed by the emperor. Their role was to defend their theme and maintain its administration. The emperor would supervise these generals to prevent any imperial decentralization. Under this system generals would recruit peasants into the army and would be payed in lands for farming. This system kept an effective military. Once the theme system became undermined by wealthy landowners, there was less incentive to serve the military and the central government received less taxes. The Byzantines also faced enemies both east and west. In the west were the Normans who in the early eleventh century expelled all Byzantine authorities in Sicily. Venetians also led the fourth crusade against the Byzantines successfully sacking Constantinople in 1204. Constantinople was taken back by the Byzantines in 1261 but this was a brutal blow to the empire. In the east the Saljuq Turks were a major threat to the Byzantines. The Byzantine forces were defeated at the Battle of Manzikert in 1071 and subsequently a civil war erupted that allowed the Turks to take most of Anatolia by the later twelfth century. This was the final blow and the Byzantine Empire simply crawled along until 1453 when the Ottomans took the capital and the small remaining land with it.

This is a pretty common question, and it's simply too broad to answer succinctly on a web forum. Keep in mind that the Byzantine era of "decline" lasted far longer than the classical Roman period.