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How to prove that the quotient group $\langle \mathbb{C},+\rangle/\langle \mathbb{Z},+\rangle$ isomorphic to the group $\langle \mathbb{C}^*,\cdot\rangle$ based on the first isomorphism theorem?

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Think about the complex exponential map $z \mapsto e^z$. It does much of what you need. From \[ e^{x + iy} = e^x(\cos y + i\sin y), \] can you determine the kernel? It isn't $\mathbf Z$, but perhaps you can scale $z$ first in order to make things work.

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Just to add a few words to Dylan's excellent answer: why would you think about complex exponentiation?

If you are trying to prove an isomorphism using the First Isomorphism Theorem, then you are trying to define a group homomophism $$f\colon \mathbb{C}\to\mathbb{C}^*$$ that sends sums to products. That is, you are looking for a function $f$ that will satisfy $$f(a+b) = f(a)f(b).$$ This functional equation should immediately suggest the standard functions that have this property: exponential functions. Once you realize that you are looking for an exponential function, it's just a matter of figuring out which exponential function you are looking for.

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