You can transplant established plants like this, however digging them up always carries a degree of risk with it, especially if you don't have a lot of experience doing it. The more roots you can dig out with it, the better off the plant will be, but that also makes it harder to move. If you end up losing some of the roots, you may want to prune the top a bit too so that the roots and the top are in proportion.
Unless there's a reason why you need it gone immediately from the spot it's in now, I would highly recommend leaving it where it is until you know where you want to move it, then transplant it from one place to the other without the intermediate step of the container. That way you only need to disturb it once and you aren't taking the risk of something bad happening to it while it's in the container (much easier to under/over water something in a container than in the ground!). But I also wouldn't recommend transplanting in the heat of summer even if you do find a place for it soon--I would wait until fall when it cools off a bit.
If you do need to put it in a container, you'll want to get rid of as much of the garden soil as you can and then plant it in a good quality, well draining potting mix. I don't think crape myrtles need anything special, but garden soil doesn't have the right type of drainage for containers so it's important to get rid of it.
You might also try posting this on the Trees/Shrubs forum--there are lots of experts over there who have more experience doing things like this than I do. This forum is more for questions about rooting cuttings, starting things from seed, etc
