Do I have enough time to do tourist things in London during a 9 hour layover at Heathrow Airport?
I will arrive at 6 am and my flight departs to the U.S. at 3:30 pm.
Also, what would be the best form of transportation to get into the city?
tl;dr Yes, 'doable' but may be stressful, will be moderately expensive, and options once in the centre will be severely restricted by the limited time available.
So, not really viable. Even without luggage you might not clear the terminal until about 07:00. There are four trains an hour (Heathrow Express) into London (and back) eg:
that take (supposedly) 15 minutes:
so you should be able to reach the centre by about 08:00-08:15. There are alternatives (as pointed out by @Moo the Heathrow Express may be as much as £37 return - though could be much less: Book in advance to save, with weekend advance tickets available from just £5.50 one way).
By Tube could be a significant saving in cost (a single is £6) but take more time that you can spare ("Journey time by Tube is under an hour and you shouldn't have to wait longer than ten minutes for a train, even off-peak." - so say 1¼ hours each way).
Hence my suggestion that for transport here "best" is Heathrow Express.
Presumably you must be back by around 14:30 even if you have your boarding pass. So must leave the centre around 13:45.
That does give you around 5½ hours in the centre but you would have to be keeping a constant watch on the time. Not for example getting engrossed in art in the Tate Gallery or exhibits in the British Museum. Maybe a bus tour (1 - 3 hours durations and frequent departures) to "cover the ground" and not further exhaust a traveller?
Add any time to get away from the terminus (and back), queue for a ticket for your preferred attraction and you may have to turn back even before having time to enter the attraction.
I am also assuming that at some point you will want something to eat and maybe go to the loo - between the two that might be another hour "lost".
In addition to pnuts answer, I would suggest a walking tour (this one features morning walks, there might be others but most I found is afternoons) because you are sitting around aplenty.