Rick,
Option 1: Stay in central London and book a car service to Heathrow. If you are there on such a short trip, it is really worth it to stay in the city.
Option 2: Spend your first 2 nights in London. Check out of your hotel on the last day and have them hold your luggage. Take the tube in the late evening (after dinner or even after a show) to Heathrow and stay at the yotel. The benefit of this is you get an extra hour of sleep since you are already at the airport. I did this once when I had a 6am flight. Yotel rooms are really small but really convenient.
Option 3: Walk, taxi, or bus (depending on where you are staying) to Paddington and Heathrow Connect train to LHR. Personally, I would not choose this option, but depending on the day of week that you are there, it is a possibility. The connect starts running around 5am. You can check the schedule at www.tfl.gov.uk and use the journey planner.
No matter what, don't stay at Heathrow for your whole trip.