If I was doing this I would take the national express coach to Stockwell Tube Station and then take the Tube to T5 via Green Park.

Not to everyone's taste but I like public transport adventures!

A Sample Itinerary:

Gatwick: 04:15
Stockwell Tube: 05:34

Stockwell Tube: 05:56
Terminal 5: 07:00


Gatwick - Stockwell (Booked through easybus) - 1.95 GBP

Stockwell - T5 - 3.10 GBP (with an oyster card) or 6.00 GBP (cash)


Or as others have suggested the direct national express coach takes 1.5 hours and is likely to cost between 20-25 GBP.