From the Civ Fanatics Colonization FAQ:
What are the requirements for
declaring independence?
The only requirement for starting a
revolution and proclaiming
independence is reaching a level of
50% for "Rebel Sentiment". Rebel
Sentiment measures production of
Liberty Bells, somewhat like Culture
in Civ4, across the empire as a whole.
At this point the European King sends the 'Royal Expeditionary Force' to stop you. You now have to fight a war of independence. As such you want to be in a position where you have enough resources (guns, horses etc.) to fight a prolonged war.
Again from the Civ Fanatics Colonization FAQ:
How can I possibly defeat the King's
forces?
...you cannot build stronger units
than the European King. Thus, you must
be patient and plan for the long-term:
since the REF comes in waves, make
sure to take advantage of lulls in
fighting to replenish your resource
stocks. Also, do not waste too many
units trying to go on the offensive --
often, it is more efficient to engage
in a war of attrition, using the
settlements' defenses and terrain
defensive bonuses to wear down the
REF. The timing of your declaration of
independence is also important. Be
certain that when you declare your
revolution, you have enough key
resources, like Tools, Guns, and
Horses, to convert colonists to
soldiers in case of an emergency.
Having a homegrown Gun industry would
also be beneficial and reduce
dependence on Europe.
