Thompson is a great trail through a scenic meadow/valley. You should know however that the 2011 Wallow Fire impacted the hills surrounding this area quite severely. It doesn't take too much away from the hike, but venturing too far up away from the river will put you into some severe fire scar.
The junction of West Fork and Thompson is called Deadman Crossing. This is a very nice area to camp and you will likely find a fire ring or two from previous visitors. But it is a "dispersed" camping area, so there are no formal sites established. If you don't find a previously used site, there is plenty of flat, grassy area to set up a tent, and even small thickets of trees for hammock camping.
On the weekend you'll likely have plenty of hikers come through. Many make Deadman Crossing the turn-around point for the Thompson Trail hike because the West Fork trail was severely burned and continuing that way isn't nearly as popular as staying down along the river.
Black bears are plentiful in the White Mountains, but are generally not problematic and afraid of people. This is especially true away from campgrounds where they sometimes become accustomed to people. If you don't feel comfortable without bear spray, feel free to bring it along, but it's not something that most backpackers in Arizona carry. Just hang your food away from camp, keep food and scented items out of your tent, and also cook away from your tent, and you should be just fine.
As with any backpacking, if you have a fire, please make sure you extinguish it completely: drown it, stir it, and feel it so it is cold to the touch before you leave it unattended. Since you'll be near water, it's also good to know that you should camp away from the stream edge, and make sure to bury human waste at least 100 feet from the river. Don't do dishes or dispose of food in the river either
Good luck, and I hope you have a great overnight!