I see a few possibilities:
1. RPM is too low - check the tension on the belt, maybe its slipping.
2. Usually these machines have a 2 PIN alternator. PIN 1 is for the idiot light and PIN 2 is the voltage sensor. The voltage sensor wire is just battery positive and the idiot light provides enough resistance to become battery positive once running. Perhaps the idiot light/resistor is faulty Without that giving the alternator it's initial magnetism, it wont work correctly. Now every alternator varies, so trace the wires before you do anything as this applies to what I worked on.
3. Parasitic drain and/or resistance somewhere in the circuit causing voltage drop. Its draining the battery and its unable to recharge after the initial crank. That's why even with a new battery you have to trickle charge it. You'll need to do a voltage drop test along the wires. Check the battery lead first.
4. The new alternator really is a faulty.

Good luck