Chinese? alternator + no voltmeter = pushing 800lbs

Please keep in mind when you read the following that I don't know you and can't assume anything about your mechanical abilities other than that you are able to remove a Valkyrie alternator, so please don't get insulted.

How do you know that the battery is dead?
How do you know that the alternator is the culprit?

Pulling the alternator shouldn't be the first step in troubleshooting "bike won't start" issues, and because a battery is new doesn't mean it's good.

How to troubleshoot this:

A. Last time riding, electrical stuff started to shut down (e.g. tachometer or electronic dash). Engine quit or wouldn't start after shutting down. Lights are dim. Putting a full charge on the battery helps until the battery runs down.

This is likely a charging system problem. This could be either alternator failure or connections/wiring between the alternator, battery, and grounding to the chassis. When the alternator fails at low mileage the problem is most likely a broken wire in/on the rotor. Some motor rewinders may be able to fix it, or you can purchase just the rotor from Honda, or you can buy a new alternator. My understanding is that OEM alternators manufactured after 2000 or 2001 have improved on this known weakness. Regardless of how it's fixed, the service manual has a procedure to bench test the alternator for continuity.

B. Nothing lights when the key is turned on, but the bike ran fine last time.

This could be the charging system, but you should still get some power out of the battery to dimly illuminate the lights, so I'm gonna say check other things first.
Did you leave the key on? Could an accessory left on have run down the battery? Has the battery been allowed to completely discharge over time, or freeze when discharged? Fully charge the battery and see if it starts.
Look at the starter relay and the 30A (or is it 35A?) fuse that plugs into the starter relay. You might find this to be a burned/melted mess requiring replacement. Another thing to check is the 55A dog-bone-shaped fusible link. The starter relay and the fusible link are under the right side cover.

C. With the key on, the lights light and the starter spins the engine (or at least you can hear the starter relay clicking), but the engine won't start. It will start when jumped from a (non-running) car. (Never jump any motorcycle from a car that has the engine running.)

This indicates a weak battery. It could be weak because of charging system problems, or because the battery is old or otherwise defective. Put a full charge on the battery, then put a voltmeter across its terminals. It should read 12.9-13.2 volts. Less than that indicates a battery that should be either load tested to see if it's good, or just replaced if it's several years old already, especially if it reads less than 12.3 volts. Start the engine, hold the RPMS at 1500 or better, and measure voltage again. You should now get a higher reading than with the engine off and a full charge, and probably something like 13.8-14.5 volts. The manual says it should be less than 15.5 volts at 5000 RPM. More than that would indicate the alternator's regulator is faulty.

D. The bike won't start even when jumped from a car.

If the starter won't turn the engine, this is either bad connections between the battery, starter relay, starter, and chassis ground, or a bad starter.

If the starter spins the engine quickly, but the bike won't start, you are probably dealing with a fuel or spark issue.

I see two have replied as I've been typing, so my apologies if I'm repeating what others have said.