From Gatwick I would take the Southern train to Victoria station ( not the more expensive Gatwick Express) then an eastbound Circle or District line underground train to Embankment station, exit out side to Villers st, just a short walk then to the road called Strand where it meets Charing Cross mainline station, Covent Garden (for the hotel) is on the north side of the Strand.
(you could also divert and walk through the gardens off Villiers st to reach Strand depending exactly where your hotel is)
Further tips: Gatwick is now within the Oyster payment boundary, load this with sufficient credit to cover the first 2-3 days including the Gatwick journey (say £40) top up later.
Gatwick fares can be varied and complicated though, a contactless bank card /pay-wave/phone-pay can be used instead of an Oystercard on the underground and buses.
Clapham Common station is on the Northern line, its worth knowing if your heading back into town for leisure from Clapham C that the Victoria line (change at Stockwell) is newer, very fast and frequent, trains every 2 minutes.
There is also a Thameslink train service between Gatwick and London, this calls at Elephant & Castle/Blackfriars/ City Thameslink/Farringdon/St Pancras, it doesn't show on the standard tube map, only the larger rail and tube map. See link
For info, i avoid using Covent Garden station, extremely crowded, small, very deep, no escalators, lifts only which can be like cattle pens.
http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf
