= Solution
Use additive notation and fix <Haar measure> $m$ on $G$. A <continuous unitary character> is a continuous homomorphism $\chi:G\to\mathbb T$. With the convention
$$
\widehat f(\chi)=\int_G f(t)\overline{\chi(t)}\,dm(t),\qquad (f*g)(t)=\int_G f(y)g(t-y)\,dm(y),
$$
the <Fubini theorem> and the character identity imply $\widehat{f*g}(\chi)=\widehat f(\chi)\widehat g(\chi)$. Also $|\widehat f(\chi)|\leq\|f\|_1$, and choosing $f=\chi a$ for a nonzero nonnegative $a\in C_c(G)$ shows that evaluation at $\chi$ is a nonzero functional. Thus each character gives a <character of an algebra> on the <L1 convolution algebra>. Here a multiplicative linear functional means a <nonzero multiplicative linear functional>; the zero functional must be excluded for the claimed bijection.
Conversely, let $\varphi$ be such a functional. It is automatically bounded with $|\varphi(f)|\leq\|f\|_1$: extend it to the <unitization of an algebra> by $\widetilde\varphi(f+z1)=\varphi(f)+z$. If $|z|>\|f\|_1$, the <Neumann series> makes $z1-f$ invertible, so its image under the unital homomorphism cannot be zero. Hence $\varphi(f)$ cannot lie outside the norm disc.
Choose $a\in L^1(G)$ with $\varphi(a)\neq0$, and let $T_xa(t)=a(t-x)$. The Abelian translation identities $(T_xa)*b=a*(T_xb)$ give
$$
\varphi(T_xa)\varphi(b)=\varphi(a)\varphi(T_xb).
$$
Therefore, defining $c(x)=\varphi(T_xa)/\varphi(a)$, one has $\varphi(T_xb)=c(x)\varphi(b)$ for every $b$. Applying this to two successive translations gives $c(x+y)=c(x)c(y)$ and $c(0)=1$. By <translation continuity in Lp on a locally compact group>, $c$ is continuous. It is uniformly bounded by $\|a\|_1/|\varphi(a)|$; applying this bound to $c(nx)=c(x)^n$ for positive and negative integers forces $|c(x)|=1$. Thus $\chi(x)=\overline{c(x)}$ is a <continuous unitary character>.
The <Bochner integral> identity $a*b=\int_G b(y)T_ya\,dm(y)$ holds in $L^1$, since $\|T_ya\|_1=\|a\|_1$. Applying the bounded functional yields
$$
\varphi(a)\varphi(b)=\varphi(a*b)=\varphi(a)\int_G b(y)c(y)\,dm(y).
$$
Cancel $\varphi(a)\neq0$ to obtain the required correspondence
$$
\boxed{\varphi(b)=\int_Gb(y)\overline{\chi(y)}\,dm(y)=\widehat b(\chi).}
$$
The recovery formula $\overline{\chi(x)}=\varphi(T_xa)/\varphi(a)$ proves uniqueness. These functionals have norm exactly one, since $b=\chi a$ with $a\geq0$ and $\int a=1$ has $\|b\|_1=\varphi(b)=1$.
For the topology, identify the <Pontryagin dual group> with these functionals and give it the <Gelfand topology>, namely pointwise convergence of $\widehat f(\chi)$ for every $f\in L^1$. If a net $\chi_\lambda$ converges to $\gamma$ uniformly on every compact subset of $G$, then for $f\in C_c(G)$,
$$
|\widehat f(\chi_\lambda)-\widehat f(\gamma)|\leq\|f\|_1\sup_{x\in\operatorname{supp}f}|\chi_\lambda(x)-\gamma(x)|\longrightarrow0.
$$
Density of $C_c$ in $L^1$ and the uniform functional norm bound one extend this convergence to every $f\in L^1$.
Conversely, suppose convergence in the <Gelfand topology> and choose $a$ with $\widehat a(\gamma)\neq0$. For compact $K\subseteq G$, the set $\{T_xa:x\in K\}$ is compact in $L^1$ by <translation continuity in Lp on a locally compact group>. Evaluation convergence is uniform on this compact set: a finite $\delta$-net reduces its supremum to finitely many convergent evaluations plus $2\delta$, using the common functional norm one. Thus
$$
\sup_{x\in K}|\widehat{T_xa}(\chi_\lambda)-\widehat{T_xa}(\gamma)|\longrightarrow0.
$$
Now $\widehat{T_xa}(\chi)=\overline{\chi(x)}\widehat a(\chi)$, and $\widehat a(\chi_\lambda)\to\widehat a(\gamma)\neq0$. Dividing this identity proves $\sup_{x\in K}|\chi_\lambda(x)-\gamma(x)|\to0$. Hence the <Gelfand topology> is exactly the <compact-open topology>, with the requested <neighbourhood basis>
$$
\boxed{U(\gamma;K,\epsilon)=\{\chi\in\widehat G:|\chi(x)-\gamma(x)|<\epsilon\text{ for every }x\in K\}.}
$$
For compact $K$, the continuous difference attains its maximum, so this strict pointwise inequality on $K$ is equivalent to a strict uniform bound. Finite intersections contain another such set by taking the union of the compact sets and the minimum of the tolerances.
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