Having the local AnimationState state = skeletonAnimation.AnimationState
is not terrible code, especially if you have a lot of AnimationState methods to call in that method.
Otherwise, organize it however you see fit. Cache the reference as a member in Start. Or don't cache it. It actually doesn't make much of a performance difference in the larger scheme of things.
But AnimationState isn't something you instantiate yourself. It's an integral part of SkeletonAnimation, which manages and updates it. You just get the reference from it. Do it only once in Start if you want.