Inside one weekers return are more frequent from Vancouver so there's my first suggestion. Although there are some from Seattle and the air is cheaper usually to there they sometime miss out on a lot of the "Inside" cruising because of time constraints.
Cruise lines is a real personal thing. Pretty well all main line major cruise lines have ships up here so you takes' your pick.
HAL, Celebrity, Princess & RCL in no particular order would be a good choice. (There are others) If you have never cruised before it really won't make much of a difference I don't think. Then you go on another line and you have something to compare. All ships do not go to all ports so if there is something you really want to do say the WhitePass and Yukon RR you have to make sure your ship goes there (Skagway)
Time to go: Early and late could be cheaper. July AUG of course more kids. My TA advised there are basically no rooms available for the rest of this month and September to Alaska and they do a lot of business.
Weather. It's the coast and hit and miss all the time. More animals moving around in Spring time. Whales take a while to get up there. So early would limit that on the other hand. Getting into the face of the glaciers is sometimes iffy and more so early season. Less fog early more later.
So best time.?? When your time, budget and itinerary all come together.
For what it's worth Princess and Holland America have been cruising to Alaska the longest.
Look for an itinerary that visits Glacier Bay...it's a highlight when cruising Alaska.
Prices are usually the best (read cheaper) at the beginning or the end of the season. I have cruised to Alaska in September. It rained every single day. That being said you risk rain in Alaska at any time of the year. The autumn colours were spectacular. I have also cruised to Alaska at the beginning of June (as I recall the ship left Seattle on June 4th). The weather was superb. In fact we got sunburned on our deck and we were able to sit out on the deck every day and enjoy it! I did miss the fall colours though.
If the budget will run to it do book yourself a balcony cabin. You will be able to step out and watch for wildlife whenever you wish....even though you may have to put on a sweater or jacket!
Pack so that you can dress in layers no matter which month you book your cruise in! When the ship is at sea it will be chilly whenever you are out on deck. Pack somekind of windbreaker/waterproof jacket "just in case"!
We did our Alaskan cruise the first part of June. We sailed on the Island Princess and really liked the ship. We chose Princess because we were intent on seeing Glacier Bay. We had a balcony cabin but were topside the whole time we were in Glacier Bay----we were dressed warmly in layers and it was cold, but we really enjoyed our time in Glacier Bay. I spent hours and hours planning our Alaskan trip and got lots of good advice from veteran cruisers here on cruise addicts. Good luck to you with your planning and ask any and all questions you have---we'll try and answer them!