"I want to become who/what I want to become" is strange in English. It's hard to say exactly why because I think I know what you mean, it could be because it's tautological (meaning it's saying the same thing twice). Okay, so if "I want to become X" then X must = a state or an object. For instance: "I want to become happy" or "I want to become a sailor". However the state "who I want to become" is vague and only restates the first part of the sentence. If I were to reword your sentence how I interpreted it it would be: "I want to become the idealized form of myself that I now hold in my head". Yet this still sounds strange because there are simpler ways of expressing a similar idea, like "I want to improve" or "I want to be a better person". Finally, I should add that in a specific context your sentence would work, for instance: Mary: I want to become a politician John: but politicians are all liars! Why do you wanna become one? Mary: what can I say? I want to become what I want to become In this context it simply restates the argument, here Mary is saying something like "I already told you I want to become a politician, and that reason alone should suffice". Last thought, using "be" instead of become seems to slightly improve the sentence: "I want to become who I want to be", or even better "I want to become the person I want to be". These sentences are all grammatical but still sound unnatural and weird to me, I think because as I mentioned above the meaning is tautological.

Who I want to become is more like what kind of person I want to be, whereas What I want to become is more like what kind of job I'd like to do