I worked there for two summers and a fall - never saw a bear. Freaked me out the first month or so. *Usually* not a lot of bears around the major areas, unless the signs go up. That aside, a hiker was killed semi close to Canyon Lodge and another killed/mauled by a bear near Lake - Elephant Rock Trail ???
The bigger concern would likely be a surprise encounter with a bison, elk or moose. I had a close call with a bull elk down in Gardiner. Zipping along on a run, not paying attention when a large "rock" stood up and I was eye to eye with a bull elk, protecting his ladies. Long story short, I survived without any extra holes in my body.
As for bears, I usually ran with others on the trails (there were a lot of other college athletes - XC, XC skiers, track etc) and we chatted it up the whole way. Also, I would run solo on the well traveled horse ride trails ... 20 - 30 horses and people on the wrangler trail 6 - 8x per day. The park folks do a good job of managing the bears. Curious/problem bears get sent to bear management areas, which are no go zones, surrounded by bear warning areas, which are surrounded by semi-safe areas. Bear management areas (hiking) really freaked me out when I saw the signs at a trail junction 6 miles from the road or site. In general, the park service does a good job of scaring the tourists out of the woods. Typically, after 400m down a real trail, you never see other people who are not park employees or hard core hiker / wildlife types.
Take warm clothes for late May, early June. I was expecting temps like 70s/40s. When I arrived at Canyon in the last week of May, there was snow on the ground and the temps weren't above 40. In Yellowstone there is the motto "there are 11 months of winter, and July". It snowed at Canyon in early June and late August.
Anyhow, it was a good time. Have fun, be safe and take a camera.