If you do the trip by public transport, the Oyster card will only cover you if you go to Aldgate by Underground all the way. You do not need to quote what zones you want to use if you are putting money on an Oystercard, just put the money on. The current Oyster single fare from zone 6 to zone 1 is £3.10, or £5.10 before 09:30 am. You can then use the Oyster for all your future travel within London, putting more money on as you need. If you are staying in London for five days or more you may save money by getting a seven-day Travelcard, which will also be given to you on an Oyster card.
The easiest way by Underground to the Aldgate area is to take a Piccadilly Line train to Hammersmith, then just walk across the platform to the District Line and take it to Tower Hill or Aldgate East. Check your hotel on a map, if it is too far to walk from either of those two stations and you must use Aldgate, then you need to change from the District service to a Circle Line train at any station between South Kensington and Monument. This will be on the same platform.
Your other public transport option, which may be faster, is to take the train (either fast but VERY expensive Heathrow Express, or slower and cheaper Heathrow Connect) to Paddington and then take a Hammersmith & City Line train to Aldgate East or Liverpool Street or a Circle Line train to Aldgate, using the Hammersmith & City and Circle platforms, not the separate District and Circle ones. You cannot use Oyster or Travelcard on the Heathrow Express or Connect trains.
Alternatively, you could share a cab for convenience, but unless you are travelling at a very quiet time (pretty much between 11pm and 6am) the Underground or train will probably be faster.