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To pay a bill for a Kindle, one usually has to pay it via the Amazon.com account to which the Kindle has been registered.
As for the debit card... any card, debit or credit, has to be registered to Amazon.com, in order to be used for payment purposes.
So to pay your bill...you need first of all, to log in to your Amazon.com account (the same one the Kindle is registered to, it has to be the same one), then either register your debit card (if it hasn't been registered with Amazon.com before... then after that), or/then go to pay your Amazon.com balance with it. (That might be under 'Your Account' in the long pull-down menu on the Amazon.com page.)
This page details what debit cards can be used:
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