First, the Roman Empire became divided between the East, under Constantinople, and the West, ruled by several different leaders. This double-reign was a new concept that weakened the empire. Infighting over leadership and decaying government strength helped set the stage for other groups, such as the Greeks and Byzantines, to become uncooperative. With it's foundations weakened, outside powers were also able to cripple the Romans.
External military threats were a major cause of Rome's fall, and its effects spread across the empire. During its days of prosperity and conquest, many of Rome's enemies were scattered tribes who lived in a small number of villages. After Rome was divided, a powerful group known as the Huns began moving west, their numbers growing with captured prisoners and new allies. People from all walks of life were eager to reap the rewards of war. They continually pressured the Roman Empire, while nations like Russia became powerful and sophisticated. What had been barbarian villages in Germany soon became 2,300 walled towns and cities. Out of these rose the countries of Denmark, Sweden, and Poland. Meanwhile, groups like the Arabs and Saracens of India and Spain sat in anger and contempt. Once its enemies united, the Roman Empire was surrounded with new competition without the proper leadership to address it.
On top of this, the Roman economy was affected by weak currency and high inflation. With all the money going towards national defense, taxes were raised to compensate. Few people actually got to enjoy Rome's prosperity. The value of Roman money plummeted to the point that bartering was preferable to paying for goods. Also, many Romans lost their jobs to cheaper slave labor, resulting in government subsidization of the working class (many workers chose to live off these subsidies, costing the government more money).
A big reason for Rome's collapse was the geographical extent of its military conquest. Rome's constant expansion required more resources and manpower to defend its borders. Additionally, conquered civilizations hated the Romans, so rebellions were constant problems. All these issues required mammoth military spending and recruitment. Manpower became so scarce that even conquered societies were allowed to join the ranks. In turn, this gave barbarians intimate knowledge of Roman battle tactics.