Tough questions! Hard to tell for certain without more of the original text. But I think....
Razor-close means extremely close. That is the two candidates are almost at the same level in the polls. We use the phrase 'razor' to mean as close as a razor to your skin when you're shaving your beard.
The second one I honestly couldn't say. The phrase 'Rinse and repeat' refers to washing your hair. You put in the shampoo, rinse it out and then shampoo again, rinse and repeat. It's possible they mean the Clinton campaign is doing the same action over and over.
Refrain in the last one means the refrain of a song. The refrain is the chorus or the line we say over and over. In this case it means Trump is repeating the same thing over and over. 'In private' is not part of the phrase. Here it just means that he is repeating this thing in private to his advisers rather than to the general public.