It may seem easy to import a character with animation but there's bugs potential with that, odd binding, joints shifted, etc.
The best way I've found to do it is:
Character Anim Import
1) Import Original un-animated Rig, cleaned up as you would clean a rig w/anim after baking
2) Create animSet
3) Double click on that to open AnimSet Editor
4) Set Skeletal Mesh to Cleaned Rig
5) File>Import FBX Animation
6) select the rigged character with animation fbx
If you don't have blendShapes you can delete all the models on the rig in the animation file and just export the baked down bind rig. So you have a character rigged, reference that into your animation scene. Animate it, save it. Run your bake script (select -hi of bind rig, bake all, delete everything that's not a joint in the bind rig or an animCurve) export that as fbx. That's what you import into the animSet, and you can import them all at once if they are in the same folder, like fbx/charName/anim/ select all an import through the animSet editor.
*Video Tutorial on this to come*