How does "do" (做) mean "become" (成)?
"做" contains the meaning of  "be ", not "become"

"我想做一个警察" means "I want to be a police"

it is "成为" that means "become", not just "成"

(成 is mostly used as a result complement for varieties of verb)

"我想成为一个警察" means "I want to become a police"


  how does the word 做 as a singular character have such a broad meaning from ‘doing’ to ‘becoming’ without the addition of another character


As I stated, 做 can mean 'be' but not 'become'

You know when does it mean 'be' or 'do' by looking into the context.
When the object after 做 is a task or action, you know 做 means 'do'; When the object after 做 is an identity, you know 做 means 'be'

Example: 

做功課，做木工，做好事 = do homework, do woodwork, do good thing (功課，木工，好事 are all nouns for action)

做學生，做木匠，做好人 = be a student, be a carpenter, be a good person (學生，木匠，好人 are all nouns for identity)

Notice: don't confuse 做(do/ be) with 造(make), e.g. 做個包子 (be a bun) is not the same as 造個包子 (make a bun)