Game engines have fairly strict rules as to what kind of aniations/rigging you can use.
You will need to have a chain of bones (if you don't care about performance, use 1 bone per segment - if you want to use fewer, the segments will probably bend)
Then parent your armature to your mesh and weightpaint your mesh (just google "Blender rigging").
Then attach your armature to your curve and animate the bones, preferrably in a way that allows looping.
Then bake the animation and you're done.