Upon finding this bay on the map, google earth and other websites I am intrigued. Only one tour I have found goes there, so it seems a bit obscure. Any comments about how best to see this village? We have a hire car and plan to lunch or afternoon tea here.

Not quite sure what you mean "how best to see Robin Hood's Bay". Since you'll have a car, you'll drive to it - either from Whitby just to the north, or from York or Scarborough to the SW or south. It's a village basically on the seacliff w/ steep-ish steps up and down the access road.

You will need to park your car at the top of the village and walk. I don't even know if you are allowed to drive there, but if so it would be quite difficult. The roads are very, very steep and narrow.

That's why I mentioned the steps -- not you don't drive into the village. You park at the top and walk down and then back up again.

You can drive down to the old part of the village (which goes right down literally to the sea) if you are delivering or if you have a property down there and need to unload (yep, we've done it as we rented a cottage down there and believe me, it's not for the faint hearted). It's a popular destination during the summer, especially at the weekend, but I like it best in the evening when the place empties.
The main street down is indeed very steep and twisting (it does have handrails in places, and steps too). Flat shoes required. Don't just stick to the main drag down, go off and explore the little lanes. There's even a tiny museum which is interesting but won't occupy you for long. Once you get right down to the sea you can have a drink in the Bay Hotel or you can walk up onto the coastal path and head either north or south along a decent but undulating path with stunning views. It's a short walk along to Boggle Hole and do some fossil collecting - the whole area features in AS Byatt's book Possession (Booker Prize winner 1990 and recommended).
www.boggle.ukfossils.co.uk
RHB has its own little online magazine called Bayfair - this is well worth a read before your visit.
www.bayfair.co.uk
A simuilar place in some ways is Staithes further up the coast. Again the idea is you leave your car at the top of the hill and walk down, although once you get down the really steep hill it does even out and some cars are allowed into the village itself. Again it's a great place to explore, and has an excellent fish restaurant called The Endeavour. If RHB appeals then I am sure Staithes would too.
www.endeavour-restaurant.co.uk

Now looking at your trip in total, I honestly don't see how you'll have time for Robin Hoods Bay. You're planning to visit Chester, York, Durham and Edinburgh in 5 days (followed by the Lake District and Peaks the next 2 days). That is about 12 days worth even at break neck speeds.

From York to Robin Hoods bay is approx 50 miles and will take probably 1 1/2 hrs. Since you will have spent much of the previous day doing Chester and then driving to York, you won't have more than a couple of hours in York if you want to make lunch on the coast. Then Robin Hoods Bay to Durham is about 65 miles and another 2 hours.

We may be well underestimating times between cities. If so, then we may miss RHB out. (We would be expecting to drive on to Durham after dark). Definitely another trip to Britain on the cards. Thanks for the advice.