Kid's geothermal power plant science project
This is more a problem of aerodynamics than an electrical problem.  When you vent a bottle of steam that way, it loses most of its energy right at the vent.

So you have to use a turbine roughly the size of the vent to get the most of the energy from the steam. It doesn't have to be such a convoluted design, a simple wheel with 8 straight flaps is sufficient. But remember, it has to be tiny.

Your DC electrical generator has a cogging torque which you had to overcome first to run from a standstill. In addition, you have to speed up against the inertia of the rotor mass. Both is going to be hard for an arrangement using improvised (non-optimized) parts.  

Again, use the tiniest DC motor/generator you can get your hands on. Go to a model railroading store, they sell such stuff and can show you motors with a small to neglible cogging torque. With such a motor, your arrangement should work.