I lived and worked in Grand Teton National Park one summer and came across bears (and a dozen other varieties of animals) here and there, but I imagine Alaska is on a whole 'nother level depending on where you are. Bears really are not out to harm humans in general. They're not something you should be terrified of (not saying you are, but a lot of people are overly cautious). Never run with headphones in, never go into the woods with a girl on her period, never go into the woods with odorous food or after just having eaten smelly meat (people have been mauled to death in their tents after getting hamburger grease on their shirts, etc). Make noises when you're in the woods if you don't want to be bothered (you can buy bear bells or the like), pay attention, and keep your distance if you ever do see a bear (or a moose, bison, mountain lion, etc). I came up on a young black bear one time and was within about six feet of it. It was as startled and scared as I was and we both took off in opposite directions.
I never ran with bear spray but probably would if I were to do it again. Just get a canister, know how to use it, and get a holster for it or something.
Any other questions? I'm kind of jealous. Alaska in the summertime sounds great to me. Spending some time there in the country is on my bucket list.