I would probably use a poly backing and a poly batting to keep it all in the same fibre group. It will be easier to wash, probably less staining and hold up to hard use.
You will need to sew carefuly and do lots of pinning as it is slippery fabric. You should be able to straight line quilt it either in the ditch or 1/4" outside the seams. Just go slow and carefully and you should have no difficulty..........and keep the hot iron away from it.
A poly/cotton blend might be good for the backing.
If it is a large quilt I might consider looking in the drapery section of the store for the backing.