Many of the legions had been decimated by a period of almost constant civil warfare from the late 4th century. As a consequence of this, and a way of dealing with pressure from barbarian confederations on their frontiers, the Roman Army had begun to depend on Barbarian foederati to fill their recruitment gap. The Roman Army had developed a Barbarian identity, which even troops from inside the Empire adopted, and was often led by generals who were of Barbarian extraction themselves (though it should be noted that they still viewed themselves, and acted as if they were Roman) such as Stilicho or Aetius, who became some of the most powerful figures within the later Roman Empire.

Due to the shortage of manpower, constant civil war between various Emperors and Usurpers, pressure from the frontiers and a decline in the ability of the Empire to efficiently extract taxes, the Empire came to depend on these Barbarian groupings within the Army. The Franks were stationed in Northern Gaul to defend the frontier against the Germanic tribes, the Goths were given Aquitaine, and later send to Spain to re-establish Roman authority and to expel the Sueves and Vandals who had settled there. The last of the legions were withdrawn from from Britain to support Constatine 'III', and usurper who attempted to use this military power base in Northern Europe to seize power for himself at a time of major crisis for the weak Emperor. Before this, the little evidence we have seems to suggest that Saxons had also been recruited into the Roman Army in Britain, and were left there to keep order and man the frontiers.

Essentially, you had an Empire already beginning to fragment, with many different local militias, who still identified as culturally Roman, but had weaker ties to Rome. The Empire became increasingly dependent on these local powers, perhaps best demonstrated by the coalition of Franks, Goths, Saxons, Burgundians and Romans who defeated Attila the Hun at the Battle of the Catalaunian Plains. As the authority of the Emperor gradually dwindled though, and Roman citizens across the Empire gradually began to realise this, these local elites, with the military power to back them up, and the ability to offer security to the local population, gradually became the new political powers in the various regions of the Western Roman Empire. Those working in administration had either already become part of the bureaucracy of the Church, which offered more secure and stable employment and a route to advancement, or they adopted a more barbarian identity and switched their support to these local elites. Others stayed loyal to Rome, or chose the wrong side, and their families gradually declined in significance.

Sidonius Apollinarius is the best individual source for how the decline of the Roman Empire affected local elites.

http://en.wikipedia.org/wiki/Sidonius_Apollinaris

Flavius Odoacer was so romanized historians aren't sure which "Barbarian" ethnic group he belonged to. Odoacer was one in a long succession of army-backed strongmen to take power, many with mixed heritage; Romulus Augustulus' himself was a puppet of his father Orestes, who was the Magister Militum and was not an ethnic Roman. In fact, the Eastern Emperor never even recognized Augustulus, which allowed Julius Nepos to rule in semi-autonomy in Dalmatia. He claimed to be Emperor of the West, and ruled Dalmatia with some autonomy.

Most of the army was made up of Foederati or ethnic "Barbarians" fighting for Rome, their allegiance to Odoacer is what allowed him to depose Romulus Augustulus in the first place. Odoacer also kept most Roman institutions intact (apart from some meddling in the court system), including the Roman senate, which he would often interact with.

The reason Odoacer "ended" the Roman empire was that he was the first to openly admit that although he might rule Italy, he was powerless beyond the alps, and didn't call himself Emperor, rather, he sent the imperial paraphernalia to Constantinople and asked to be recognized as "Consul", keeping all of the functions of the Roman Administration in Italy intact. He initially played lip service to Julius Nepos, minting coinage in his name, but may have been involved in the plot whereby he was murdered. After the murder of Julius Nepos, Odoacer asked to be given the title of Patrician by the Eastern Emperor, and be recognized as the ruler of Italy under the wing of a single Roman Empire.

At this point, the Eastern Emperor Zeno, had his own issues to deal with involving "Barbarian Generals", namely involving two Ostrogoths, both confusingly named Theodoric (this in addition to social, religious, and political unrest in his own half of the empire). Although they had been given lands in Pannonia in exchange for the defense of the upper Danube, Zeno came to the conclusion that the Ostrogoths were more trouble than they were worth, and convinced them to go conquer Italy, so that they both got out of his way, and got rid of Odoacer.

Theodoric the Amal, king of the Ostrogoths had been raised in Constantinople, his people were largely Romanized and had been living within the Eastern Empire's borders for centuries. When he defeated Odoacer and set up the Kingdom of Italy, he did his utmost to further maintain the Roman institutions, law, and judicial system. (I suggest this book every time Theodoric comes up, he was awesome: Theoderic in Italy by J. Moorhead (Oxford University Press, Oxford, 1992)

Indeed, you could say that the "Ancient Roman" administration continued to function, with varying degrees of efficiency, up through the Lombard rule in Italy, as was the case all through Europe, in the Visigothic and Vandalic kingdoms, most of the actual administration was done by a Latin-Speaking elite (and most of the "Barbaric" ruling class was very romanized). Heck, In Italy, I suppose you could draw a line of administrative continuity up to the Communal Era; or judging by how long it took me to register to vote a few years ago, up to the modern day.

edit: typos