Where to report payments made to C Corporation?

It's the same as other business expenses. Schedule C Part II is all about expenses. Pick the appropriate categories, and if an expense doesn't fit one of the categories they provide put them in Part V (Other Expenses). Aggregate payments for like-types of expenses in the Other section if using it.

As for proving the expense, it's easiest if you have separate personal and business accounts, so that all business ins and outs hit one account. Regardless of whether or not you have separate accounts, it's important to keep good records of business income/expenses. Preserve invoices/bills/receipts and maintain a ledger that shows the same with dates paid/received.