How to politely decline an interview for a PhD program?

They seem to have all the facts already. Just tell them that you had already made travel arrangements to visit A and B on the dates that C wanted you to come out, and that you have to regretfully decline to come to C. They're not going to be mad at you.

You have asked for an alternative interview date, gave them a list of options where you're available, and made efforts to reschedule conflicting appointments. I don't think there's anything more that could be expected of you, so it should not be perceived as impolite if you just inform University C that unfortunately you can't come to any of the two offered interview dates due to conflicting appointments. If you want, you can explicitly mention interviews at other universities - that will make it clear that you're genuinely interested in doing a PhD, and it will be understandable to most academics that everybody has a ranking of opportunities in such situations.

You could offer to retract your application, or state that you expect that they won't consider your application any further, but I don't think any of these would be required.

I don't see any reason they would be upset by you telling the truth. In fact, trying to get too deep in explaining "why I am choosing to go to A and B instead of you" could potentially be more "offensive", if that's what you're worried about. Concerning politeness, I would call if possible; quickly, too. There's no reason to leave them waiting to hear back from you if the answer is 'no'!

Be short, sweet, and to the point.

"I am sorry to say that I won't be able to attend an interview on either of the dates provided. If a time comes up other than those provided, please let me know. I apologize for any inconvenience."

It's also possible to leave an opening for a possible interview with University C later that way.