I want to pay my iCloud bill

You can use an iTunes gift card to pay for iCloud, if you don't have any other payment method for your Apple ID available:

What you can buy with your Apple Gift Card or Apple ID balance - Apple Support

Payment methods that you can use with your Apple ID - Apple Support

You should contact Apple Support about that. Only they can look into your personal settings from their side, to find the issue and help you solving it.

Apple Support Phone numbers

I've updated payment information but Apple keeps sending messages that "there is a problem with your payment information."