"Enchantments in May? Depending on the experience you're hoping for, this area is a good 2 months away from being ready for good hiking. Even with the insanely warm weather we're having (loving it!), there are still likely many feet of snow up there."
I've done the full traverse in May twice, starting from the Snow Lake TH. Both times there was a lot of snow, as in up to our thighs most of the way thru the Lower Enchantments on up almost to Aasgard Pass. The second time, the upper 300 feet vertical on Aasgard Pass was blue ice and it was a real butt puckering experience negotiating it. We had decided not to bring crampons, based on our experience the previous May, BIG mistake. We got down safely, but it could easily have turned out otherwise. It is doable, but I would strongly advise you to come well equipped for serious conditions and be in top shape. It is a very strenuous undertaking in May. With hindsight, skis would probably work better than snowshoes(we took neither), but you'd have to carry them a long way on either side of the actual Enchantments section of the hike. I would concur with Curt's recommendation to consider waiting for the snow to clear, especially on your first attempt. It is a gorgeous area, and there is little to gain from turning your first attempt into a survival march.
A good May alternative would be a hike up the Fourth of July Trail up onto Icicle Ridge. Once up on the ridge, there are two excellent options: either turn west and hike up to the remnants of the fire tower, or turn east, if you have snowshoes and run the ridge down until you pick up the trail that leads down to the TH just outside Leavenworth(this option would require pre-positioning a car or bike there, or arranging a shuttle back to your car at the 4th of July TH). The snowshoe trip is about 12-13 miles of ascent, rolling terrain, and the descent on trail, and is a PNW classic spring trip. Either way, you are looking at nearly 5000' of elevation gain, and some magnificent views, so bring a camera. Bonus: It'll put you in great shape for The Enchantments Traverse in July. :)