The train is the fastest way,
Spot on...Heathrow Express ("HEX") is the way to go. Exit T5 arrivals, turn right and the lifts to Trains are on your left towards the end of the building( but do
not get on the Underground by mistake).
T5-T3 in 15 - 20 minutes, much less if you get lucky with timings and ....it's free.