"should I just link my credit card through Banking and enter each expense?"

Linked or not, the entry of an expense paid by Credit Card is similar to one paid from Checking (as paper or debit card); the difference is the Source of Funds. You didn't use your own money; you Charged it on that card.

"Or should I just enter the total payment to the credit card as a debt payment?"

The Payment is a Transfer, between checking and the credit card account. That way, they Both Go Down.

"Or do I do both?"

The first is the spending of someone else's funds by using their credit card they provided to you.

The second is your Cash Flow as an expediture but not as an expense account entry.