It's entirely up to you how billing should work. Some Partners do not want the hassle of billing customers. In those cases, when you upgrade the site you can choose for Adobe Business Catalyst to have the billing relationship with your customer. In this scenario your customer will be aware of Adobe's role since the customer will receive invoices with the Adobe brand on it, charged at the scheduled rate. You will need to obtain the customer's credit card details before upgrading the customer from a trial site to a paid site.
Alternatively, you can have the billing relationship with your customer. You will be billed at the scheduled rate for each site and will be solely responsible for any billing and collection of payments between you and your customer. Premium partners will have access to some great tools to help manage customer billing, including the ability to set up consolidated billing for your customer sites.