My cat had 5 kitten this morning. But where she had them is very messy. Should I move her and the kittens to a cleaner bed?

When can I change the bed the kittens were born on.

You can change the bedding now if you want, just be gentle. Move them out of the way into a new warm area, change the bedding, and move mom and the kittens back to the original spot.

May I asked why your cat wasn't spayed?

I would consider placing another box with clean bedding next to the one you have now.
Cats can be pretty touchy about you handling their new kittens at first so I would give her the option of moving them herself first.

We did this with the stray that had kittens under our house. She was fairly tame as my mom had been feeding and handling her for quite some time but we felt it was better to give her the option of moving the kittens herself.
We placed the new box right next to the old one... came back an hour later and she had moved them all.
After the first day or two we started petting the kittens while the mom cat was there and she didn't mind at all... that was when we knew it was OK for us to pick them up and change to bedding ourselves.