When fishing upstream nymphs, I've seen advice to move the fly a little faster than the flow. I've always just left a little slack in the line and let the fly drift but my hit rate is poor.

Any thoughts, especially if fishing with a nymph and a dry on a dropper?

No different to any other kind of nymphing... you need to remove the slack from the line and retrieve the line so as to stay in touch with the flies. Use the "high stick" technique, a long rod will come in handy for this. Dead drift with upstream nymphs is going to be unproductive unless you can pick up the take.