Those things bonnielady brought for formal nights will work on Royal Caribbean too. If you want to avoid dressing up altogether, you will have to eat in the buffet area on formal nights. Jeans are not allowed in the dining room at dinnertime (ok for breakfast and lunch, though.)
Wear layers in Alaska, as the temperatures could be anywhere from 50s to 70s, possibly even low 80s. Expect rain (and if you get sunny days, it will feel like a bonus!
) Bring binoculars too, if you can (small ones will travel easier). Have a great time!