Use the bill's date for the bill and the check date for the check.
When you write a check, the expense is recognized on a cash bases based on the check date, not when it clears your bank. Otherwise you'd be subject to the whims of those you write checks to, when - if ever - the decide to deposit the check.
The same is true for Payments you receive from customers. The income is recognized on the date of record for the payment (perhaps the day you received it) and not when you get around to going to the bank and depositing it, which is not relevant.
