Besides The History of the Decline and Fall of the Roman Empire by Edward Gibbon, you can rather start off with the still fundamental Geschichte der Stadt Rom im Mittelalter by Ferdinand Gregorovius, published between 1859 and 1872. You’ll easily find it translated into English (History of Rome in the Middle Ages), a full text here. In the volume I, book II, you’ll find some information dealing with the beginning of the reign of Flavius Odoacer and the period between the 5th and 6th centuries AD.
Alternatively, you can check out the third volume of the “Storia di Roma” directed by A. Schiavone, entitled “L’età tardoantica. Crisi e trasformazioni” (Torino, 1993) and the bibliography cited there.
MR