Note that the group ring $\mathbb C[\mathbb Z]$ embeds into $\ell^1(\mathbb Z)$ as the subring of finitely supported sequences. Furthermore, its image is dense.
Thus a continuous homomorphism $\ell^1(\mathbb Z) \to \mathbb C$ is determined by its restriction to $\mathbb C[\mathbb Z]$, and so it might help to determine
the homomorphisms $\mathbb C[\mathbb Z] \to \mathbb C$ first.
[Aside: Actually, all homomorphisms $\ell^1(\mathbb Z) \to \mathbb C$ are continuous, because maximal ideals are automatically closed; but it is helpful to remember the continuity property, since it allows one to interpolate from the pure algebra of $\mathbb C[\mathbb Z]$ to the analytic context of $\ell^1$.]
Now these latter homomorphisms are determined by where the element $1 \in \mathbb Z$ goes; once we know that, everything else is determined by the homomorphism
condition. (In more representation-theoretic terms, homomorphisms $\mathbb C[\mathbb Z] \to \mathbb C$ are the same as one-dimensional representations,
or characters of $\mathbb Z$, and these are determined by their value on the generator $1$ of $\mathbb Z$.)
Thus the homomorphisms $\mathbb C[\mathbb Z] \to \mathbb C$ are in correspondence with the possible images of $1 \in \mathbb Z$, which can be any
non-zero complex number, say $z \in \mathbb C \setminus \{0\}.$
But not all of these will extend to continuous homomorphisms $\ell^1(\mathbb Z) \to \mathbb C$.
Slightly informally, any such homomorphism would (by continuity) have
to map the sequence $(a_n)_{n \in \mathbb Z}$ to the complex number
$\sum_n a_n z^n$, but this latter series won't converge in general.
Note that since $(a_n)$ is assumed to be in $\ell^1$, this series will converge if $|z| = 1$; this shows you that of $|z| = 1$, our homomorphism
does extend from $\mathbb C[\mathbb Z]$ to $\ell^1$. A little more argument
shows that if $|z| \neq 1$, then there is no such continuous extension.
General philosophy: think of $\ell^1$ of a group as a completion of the group
ring. First see what you can work out by pure algebra (manipulating the group ring), and then see what additional constraints the analytic conditions impose.