Synthesis of hydrazine with bleach and ammonia
I sympathize with your desire to launch an action figure. However, the chemistry you propose is really dangerous. You would be better off with black-powder based rockets.


  Is there any way to synthesize hydrazine with bleach and ammonia?


Do not do this. 

My answer is not "yes" or "no". My answer is Don't. Do not try this without proper safety equipment and training, and then think twice or three times before doing so, and then have an ambulance standing by. Certainly don't do it at home, or by yourself.

This particular reaction is capable of producing hydrazine. However, it also producing a whole host of additional stuff, like chloramine, hydroxylamine, nitrogen trichloride, hydrogen chloride, and chlorine gas, all of which are either gases or volatile liquids, and all of which are poisonous. Hydrazine is poisonous. Nitrogen trichloride is a shock-sensitive explosive on the order of a sneeze can set it off. The reactions are as follows.

Chloramine:
$$\ce{NH3 + HOCl -&gt; Cl-NH2 + H2O}$$

Dichloramine: 
$$\ce{Cl-NH2 + HOCl -&gt; NHCl2 + H2O}$$

Nitrogen trichloride:
$$\ce{NHCl2 + HOCl -&gt; NCl3 + H2O}$$

Hydrazine and HCl:
$$\ce{NH3 + NH2Cl -&gt; H2N-NH2 + HCl}$$

Hydroxylamine and HCl
$$\ce{NH2Cl + H2O -&gt; NH2OH + HCl}$$

Chlorine gas:
$$\ce{HCl + HOCl -&gt; H2O + Cl2}$$

Any organic impurities in the mix can lead to other deadly stuff being produced.

This reaction is not controllable to produce a single specific product. Everything it produces is poisonous and/or explosive. I will recommend against, and I will not help you find a recipe.