PART 1

When and how did the Dutch come to consider themselves (or be considered) a separate people from the Germans?

Though I think your question is perfectly understandable from a modern perspective, it's important to point out beforehand that it is based on a false premise: namely that a German ethnicity, nation or group identity predates that of the Dutch.

It does not.

Common origin and Germanic identity

Both the modern Dutch and Germans (and English, and Scandinavians) originated from a common cultural group, known as the Germanic tribes, who spoke various related Germanic dialects.

It would be wrong to think of this group as a modern nation or ethnicity: Germanic society was clan-based and tribal. They spoke related dialects and worshiped roughly similar deities, but other than that; there was little love lost between them. Their society, especially before the formation of large confederations following Roman contact, seems to have been one of near constant, low intensity, internecine warfare. Today, historians call these people 'Germanic' but this was not an endonym used by the actual Germanic tribes. It originated as a Roman catch all term for the peoples living east of the of the river Rhine, most likely taken from the name of a single Germanic tribe which was then applied to a multitude of different tribes. Some of these tribes were even most likely Celtic-speaking, others (like the Goths) were undeniably Germanic-speaking, but were never referred to as being a Germanic tribe by the Romans. As far as modern historians and historical linguists are aware, the Germanic tribes had no terms equivalent to 'Germanic' or 'Germanic people' as used today.

Medieval class system and the reforming of identities

Following the Migration Period and the collapse of the Roman Empire, Germanic customs and the Roman legacy merged to form a societal system which would later be called feudalism; in which society was divided into three classes: a warrior nobility, the clergy and peasants. By the 9th century CE, what remained of the individual tribal identities was lost as someones identity was no longer based on a form of extended kinship, but on social class.

Nationality and ethnicity, did not exist in any modern sense. This does not mean that cultural, religious or linguistic differences between regions did not exist, they most certainly did; but they did not result in a feeling of unity.

To give a concrete example:

• A German officer of the Prussian Junker gentry and a commoner German soldier in the late 19th century; would agree that, despite their differences in social standing, education or wealth; they were fellow Germans, on the basis of speaking the same language and living in the same country.
• To a medieval German duke, the fact that he could speak the same language and lived in the same duchy as one of his peasants would have been almost insignificant as far as his personal identity went. This duke would have identified himself far more closely with members of neighboring French/Polish/Dutch/Spanish/English-speaking nobility than with his lowborn peasant.

(SEE PART 2 FOR THE REST OF THIS ANSWER)

Summary / TLDR

The Dutch were not, nor did they ever consider themselves to be Germans in an ethnic/national sense. The establishment of the Dutch nation predates the German one by several centuries.

Sources:

• Ton Derks: Ethnic Constructs in Antiquity: The Role of Power and Tradition, Amsterdam University Press, 2009
• Was ist deutsch? Fragen zum Selbstverständnis einer grübelnden Nation. Nürnberg, Verlag des Germanischen Nationalmuseums, 2006.
• Peter Wiesinger: ‚Nation‘ und ‚Sprache‘ in Österreich, de Gruyter, Berlin/New York, 2000
• Rogers Brubaker: Citizenship and Nationhood in France and Germany, Harvard University Press, Cambridge,1992.
• Bernd Schneidmüller: Reich – Volk – Nation, Harsowitz, Wiesbaden 1995.