Fuel, Ignition, Air. We should be able to rule out air and when it dies, does it sound like it's starving? Can you manually rev it to see that you are getting fuel? If it just stops, I would lokk at your ignition source. Just my .02
Upon cold start, push the throttle all the way to the floor. Do the choke flaps close all the way?
If it's an electric choke, check for power at the connection to the carb when the engine is running.
What should happen is the choke should close all the way when you floor the throttle before starting. Upon startup, the choke should open only part way. This is called a "choke pull-off", and it should only open the flaps about 1/8 - 3/16" initially. As the engine warms, the choke should gradually open, until the engine reaches operating temps, at which point the choke should be fully open.
It may, or may not, be your problem. A faulty choke will make cold starts an adventure, at best.
Usually, you can overcome a bad choke by massaging the throttle while the engine warms. That's assuming the choke isn't stuck closed. If it's stuck closed, then your engine may fire briefly, but it will quickly flood and die.