How to catch a train from Gatwick to London- Bayswater area hotel?

Follow the signs from the terminal to the train station at Gatwick. Buy your ticket at the station. Then take a Southern train to Victoria. Buy an Oyster card for GBP5, top it up with a few pounds then take the Tube on the District line to Bayswater. Walk to hotel.

When you are staying anywhere or going anywhere in a large European city. One of the first things to know about your hotel/destination, at least as important as their street address, is the nearest underground (or métro, or U-Bahn, or ...) station. If you look on the hotel website for the Grand Royale Hyde Park hotel it shows it is equidistant between Bayswater and Queensway stations.
If you then look at the London tube (Underground) map here: http://content.tfl.gov.uk/standard-tube-map.pdf
You can see Bayswater station in square C3, and Victoria station in square D4, connected by the yellow circle line.

It is then easy. DO NOT buy a pass.. As Ramblin' on says, buy a ticket from Gatwick to London Victoria (Gatwick station is part of the South Terminal). If you arrive at the other terminal, you need to get the inter-terminal shuttle.
At Victoria, follows signs to the underground, buy two Oyster cards (one each) and load them up with cash. These will be your most useful purchase for your whole trip. Follow signs to the circle line (yellow), catch a train (in the correct direction) and get off at the 6th stop (Bayswater) - EASY.

The last week in December starts with Christmas Day and ends with New Years Eve.

Since transportation is somewhat sparse that week, including NO trains, buses or tubes on Christmas Day, a special schedule for Boxing Day, and a bit of a madhouse on New Years Eve, it would be good to have a better idea of exactly which days you have planned for arrival (from where?) and any departure you would be planning before the 2nd of January.

Regarding using the tube from Victoria, how much luggage are you intending to bring? Have you joined the Rick Steves only bring what you can carry club, or are we talking suitcases? Any mobility problems?

yeah should be the Circle line rather than the District

Let me reiterate Nigel's point that THERE IS NO PUBLIC TRANSPORT SERVICE AT ALL IN LONDON ON CHRISTMAS DAY and may not be any to Gatwick Airport on Boxing Day. Every year we get people on this board who bought air tickets to London on Christmas Day and are shocked to discover that "a world-class city can just shut down on Christmas Day". YES IT DOES. If you planned to travel internationally on either day, get as much of a refund as you can NOW and buy new tickets.

Thanks for the great advice!! We arrive on the 29th so no problems with Christmas day shut downs. Will New Year's day also be a shut down day?

January 1 is not a shut down day, but a light timetable and check for any engineering works.