No stable method of succession. Often when an emperor died, the top generals would fight it out to see who would be the new emperor. This would plunge the empire into frequent, ruinous civil wars.
Overrelliance of the military. The military had way too much power in Rome. New emperors would have to bribe the soldiers to keep them loyal, which drained the treasury. When the soldiers weren't happy they'd often revolt and kill the emperor. Sometimes they'd go on a rampage and pillage from their own citizens.
Problems with barbarians. Mass migrations in Asia down through northern Europe pushed barbarians into Roman territory. The result was frequent barbarian raids. This stretched the Roman military thin. Often Rome bungled negotiations that could've settled problems peacefully or turned them into allies. Eventually Rome came to rely on barbarians as mercenaries, and they often had loyalties other than Rome. Towards the end Rome had no money to pay mercenaries, so instead they'd grant barbarians land in exchange for military services. The net result was that Roman territory shrank, along with its tax base. This was most responsible for the final collapse of of the Western Empire.
Overexpansion. As Rome was expanding, new conquests were a source of wealth, slaves and new land, all which economically benefited the empire. Once it stopped growing though, it was left with large borders that needed to be defended, and this turned into a financial drain instead of a financial benefit.
Lack of money. Roman mines eventually started to run low on gold and silver. So they debased their coins, which resulted in inflation and other economic problems.
The splitting of the empire. This started as a benefit, since it made it easier to defend the empire. The problem was most of the wealth was generated in the Eastern half, leaving the Western half with economic problems.
Plague. At the end of the Pax Romana a severe plague killed off many citizens, depriving Rome of potential soldiers and tax revenue. Oh and sorry no one wanted to help you when you needed it... It happens to me too...
