Welcome to Alaska. I'll try and throw a few ideas at you here: So much depends upon how much time you have for your trip, you didn't mention which Radiance cruise you would be using. However, after looking at their itineraries for this vessel it looks to me like you are going to have a great cruise. You get Hubbard Glacier, I believe, too, which I find to be one of the best nowadays for calving viewing. You will be on both the Alaska Inside Passage and the Canadian Inside Passage, which is good.
I'm assuming you are coming on up to central Alaska but not doing a land portion afterward. In the event you will have time for a land portion, do you know about "Cruise Tours"? I'm a fan of cruise tours but a lot of folk are not, they rather do the land portion on their own. Either way is great, but I suggest you get a copy of "The Milepost", fun to read and to study, most importantly it will give you a good idea of what you may prefer with respect to shore excursions in both Alaska and Canada.
About 65% of cruisers to Alaska only do the Southeast (Panhandle) portion of the State. For those who are on of these 7 day (round robin) cruises they probably won’t be crossing the Gulf of Alaska and traveling further north to South Central Alaska or going on into the interior on a land tour, but most likely doing a cruise of Southeast Alaska only, originating and ending in either Vancouver or Seattle. Almost all of these cruises do the "Alaska Inside Passage".
My personal preference for these round robins is to get on a cruise that voyages through the "Canadian Inside Passage" as well as the "Alaska Inside Passage". Meaning, they travel to the east of Vancouver Island as opposed to the west of the island, west of the island places the vessel out into the North Pacific where scenic voyaging is limited to non-existent, rougher water there too, sometimes. Vancouver is the best port for this. (Note: this is also important even when you are going to cross the Gulf of Alaska and terminate or originate in Central Alaska, cruising east of Vancouver Island is a must on at least one segment).
Beware when the line or your agent simply says you are going to be in the "Inside Passage". Being life long Alaskans we have taken only one cruise to Alaska, which was on Regent, a two weeker, originating in San Francisco and ending in Vancouver, around the end of May and first of June. Hot weather even, 72 F, for Alaska that is. We sailed as far north as Valdez in Prince William Sound and hit every obligatory port in Southeast. Excellent cruise; the only downer was, both north and south, we skipped the Canadian Inside Passage and traveled to the west of Vancouver Island. Some Canadians on board weren't very happy in that they had been led to believe they would be cruising on the east side of Vancouver island.
I'm not certain when you are coming either. For early June you shouldn't have to pack any different than you would for a Baltic or a New England cruise that time of year. In late June many of the younger visitors in central Alaska are wearing outdoor shorts, it can hit over 90 F during the days then in Central Alaska.
Since you might get into a stretch of fairly warm weather at times dressing in layers is appropriate. The mosquitoes are pretty much not a problem in Southeast Alaska unless you get away from the coast and in Central Alaska, in early June, not a problem either, usually. However, in late June, watch out, so don't forget the bug dope, treat the mosquitos with respect, as they are our state bird, LOL. Mosquitoes are usually on the wane by mid August.
You might run into a bit of rain, particularly on a shore excursion in Southeast, so like us bring along fold-up light wind breaker rain jackets. Umbrellas more often than not are a joke because if you have rain you usually have wind that destroys them. I prefer some real light fold up rain pants, which you probably will never use but if you need them they really are nice. For shoes, I always bring my running shoes and if we are going to get into wet terrain or do some trail walking or climbing I bring my lighter waterproof hiking boots (note: hiking boots not hiking shoes).
A light sweater is great but even better is a hooded sweat shirt and a couple of baseball caps. Bring binoculars. Light all night pretty much so no need for flashlights. Sun glasses you might need, especially around the glaciers, but I wouldn't worry about sun screen unless you plan on glacier hikes or some mountain climbing above snow line.
I was born and raised on Prince William Sound, when I was a kid about the only practical way for the family to visit relatives in the states was to travel on the Alaska Steamship Company’s vessels, which were combined passenger and freight, to and from Seattle. From Cape Spencer (the northern tip of Southeast before the vessels leave, or enter, the Alaska Inside Passage) Alaska Steam always stayed out of the Gulf and voyaged all the way to Seattle via the Inside Passages. I’ll never forget the scenic wonders of all that.