Heathrow Airport to Chipping Campden

Train to Moreton in Marsh then Stagecoach bus 1/1a/2/2a. In combination those buses run roughly every 90 minutes from MiM railway station.

Since last weekend that service also runs on a Sunday with 6 departures.

Most buses are headed for Stratford on Avon.

The various rail routes to Moreton in Marsh are well rehearsed on here.

The "best" way in my book is a private car service. There are many of them and I'd be wary of any company that self promotes on this website as it violates the rules. Of course, this will cost a lot more than the train and a bus.

Blackberry Cars and Just Airports are two companies mentioned frequently here.

Thanks for the recommendations! I'm flying in on a Thursday and was planning on starting the hike on Friday. But now I wonder if I should add a buffer day?

Assuming you are arriving early morning as most TA flights do, arriving Thursday and starting Friday should be OK.
Heathrow to Moreton in Marsh is around 2 hours on the train depending on connections/route then around a 45 minute bus ride.
So you should be there in time for lunch. Have a snooze, familiarise yourself with the town, have an early night.
Start next day.
Now of course the great unknown is a flight cancellation or heavy delay.

With a buffer day that would give the option of a pre day in Oxford, Stratford or Cheltenham (on the 606 bus) if all went well.