How did Native Americans react towards the fighting between Britain and France in North America during the years 1756-1763?

The period you specify is the French and Indian War, also
known as the Seven Years War. Native Americans fought on either side of the war, not so
much because they agreed with or liked the Europeans with whom they fought; but because
they did not like and were at war with the Native Americans on the other side. A little
known fact is that the war in America actually began as a conflict between two Indian
tribes.

Samuel de Champlain, the French explorer, had
intervened in a dispute between the Huron and Iroquois Indians. Champlain fired into the
Iroquois and killed two chieftains; as a result of which the Iroquois had no further
dealings with the French. The Iroquois traded with the English and actually provided a
buffer between French North America and the English colonies. The Hurons reflexively
allied with the French.

In Spring, 1754, George Washington,
then a twenty eight year old colonial in the British army, was sent to warn away French
troops who had trespassed on land claimed by the British. A brief skirmish occurred, and
a truce was called to discuss a resolution. When Washington turned his head, an Iroquois
who was with him split the skull of the French commander and washed his hands in the
man's brains; not that he had a quarrel with the Frenchmen; but rather because the
French were allied with the Hurons, the Iroquois' hated enemies. The end result was a
much larger battle and the opening salvoes of the Seven Years War in
America.

Although the War technically ended in 1759 with
the Battle of Quebec, there was still fighting for an additional four years as the
Indians refused to stop. Pontiac led a group of Ottowa Indians on the side of the French
in hopes of bringing about a pan-Indian revolt. In South Carolina, Cherokee Indians
managed to push the frontier back 100 miles. The War ended with the Peace of Paris of
1763.

An indication of the importance of Indian relations
is demonstrated by the Royal Proclamation of 1763 issued after the Peace of Paris. The
Proclamation forbade English settlement west of the crest of the Appalachian Mountains
in hopes of maintaining peaceful relations with the Indians who occupied that
area.

An excellent source of further information is Fred
Thompson's Crucible of War, which explains the origins and course
of the Seven Years War and its relationship to the American
Revolution.