Engine needs the following to run - fuel, air, sufficient compression, decent spark (a big fat spark not some piddly arsed thing you can hardly see) at the correct time. Remember that it is much harder for the coil to produce a spark in the cylinder under compression than when you just have a plug attached to the lead and held against the engine (not with your bare hands 
).Assuming you have fuel getting to the cylinders - you'll know by now by the smell if you've been winding the engine over so much! No smell? I'd undo one of the drain plugs a little on one of the fuel bowls and see if there is any fuel in the carbs. If this is a yes next place to look is the spark plugs I'm afraid.
Plugs all wet with fuel? They could be fouled! If no fuel check that it runs from the tank when the tap is at prime position, if yes next look at the fuel pump, this is vacuum operated and obviously needs a correctly connected pipe from the inlet manifold to work (as does the fuel tap). This all check out 
Have you got the ignition coils connected to the correct spark plugs?
I assume the starter motor is winding the engine over strongly on a fully charged battery? It is not unknown, especially on big hairy chested twins like the TRX that take a lot of starter power to turn them over to strike a problem called "coil robbing" where the starter motor takes all of the current and leaves insufficient for the coils to fire the plugs. This often happens if a cell in the battery is starting to die.
Will be interested to find out what the problem is
Keep us posted 
