If you're hiking on the south side, there is a nice little FS CG just below Timberline Lodge, Alpine Campground. Check the Zigzag ranger district for info, it usually opens in mid-June and closes just after Labor Day.
You can also dispersed tent camp outside of the Timberline permit area, which is a fairly narrow strip running parallel to the chair lifts. The Timberline website will show the boundaries and once you're outside of that no one is going to bother you.
Don't know what your level of backcountry experience is, but there is also a series of tent sites cut into the snow on the hogback at 9200 ft. While this is intended as an overnight area for summit climbers, plenty of alpine travelers make this their destination, and camp out in a magnificent open area--as long as the weather is perfect. This is also an area that has claimed many unprepared lives, so know your ability level and equipment.
Not to be a spoilsport, but if you camp above treeline you're either camping in the snow or in the dust. Dust dust dust. Even where there's still enough snow to ski to the parking lot, if there's a breeze on there's enough dust to make your teeth gritty.
Yes, there are bears lower down the mountain (closer to Government Camp), as well as deer and I believe elk. Funny thing is, I've never seen a coyote in the region and I thought they were all over North America. All wildlife just wants to be left alone, and bears are willing and able to enforce that.
Good luck!