has regular coach service (about every 10 minutes!!) from Gatwick Airport to Heathrow Airport. The journey is about an hour and fifteen minutes or so, and the price will be somewhere between 15-20 GBP depending on the day & time. You can get the Go Anywhere (read: cheap) fare if you book the journey during the month you'll be taking the trip.
You might also be able to find a train fare for a slightly comparable price, but I think it takes a little bit longer. Check here
for timetables & prices. Good luck!