Wonderful deals! Too bad my kids are all >11 and can't travel on those dates. I think the cruise the OP is looking at is August 22, 2014 on Millie. An interior for four (two kids under 12) is $ 1,038. BUT, Celebrity's website doesn't add port fees and taxes until you hit the BOOK button, so the total is really $ 1,753. Still an incredible deal when you consider DCL is well over $5,000.
OP, please remember your kids will need passports. And PLEASE look into the Alaska Rairoad, assuming you will fly into Anchorage. It takes you down to Seward very early the morning of the cruise, has a glass domed top, a dining car, etc. Really a magical way to see Alaska before getting on the ship. You will then have time to go to the Sea Life Center in Seward before boarding the ship. Have fun whatever you choose!