How can I look at my character?

Solution 1:

If rotating the camera does not suit your needs, you can get a close-up of your character's face by visiting a barber or plastic surgeon. I don't have a comprehensive list of their locations on hand, but I do know there is a plastic surgeon in Diamond City.

I've heard that Codsworth can give you a haircut as well, but I have not confirmed this.

Solution 2:

If you put your weapon away (long press of the reload button), then rotating the right stick in 3rd person will be in freelook mode.

Solution 3:

On the PC you can hold down MOUSE 3 (middle mouse) while in third person mode to activate free look, allowing you to rotate around your character.

I'm unsure of the equivalent controls on console.