= Solution
For a <divisor on an algebraic curve> $D=\sum n_PP$, define its <line bundle associated to a divisor> inside the <sheaf> of rational functions by
$$
\mathcal O_X(D)(U)=
\{f\in k(X)^*:v_P(f)+n_P\ge0\text{ for every }P\in U\}\cup\{0\}.
$$
Locally choose a rational equation $f_i$ for $D$, as in the construction of the quotient <sheaf> above. Then
$$
\mathcal O_X(D)|_{U_i}=f_i^{-1}\mathcal O_{U_i},
\qquad
\mathcal O_X(D)_P=t_P^{-n_P}\mathcal O_{X,P}.
$$
Consequently this is an <invertible sheaf>. Multiplication gives an <isomorphism>
$$
\mathcal O_X(D)\otimes_{\mathcal O_X}\mathcal O_X(E)
\cong\mathcal O_X(D+E),
$$
as is checked on the displayed local generators. If $D=\operatorname{div}(f)$, multiplication by $f$ identifies $\mathcal O_X(D)$ with $\mathcal O_X$. Thus $D\mapsto[\mathcal O_X(D)]$ induces a <group homomorphism> from the <divisor class group> to the <Picard group>.
For surjectivity, let $\mathcal L$ be an <invertible sheaf>. Trivialize its one-dimensional rational fibre over the <function field> $k(X)$; equivalently, choose a nonzero rational section and identify $\mathcal L$ with a subsheaf of the rational-function <sheaf>. A finite trivializing cover gives
$$
\mathcal L|_{U_i}=a_i\mathcal O_{U_i},\qquad a_i\in k(X)^*.
$$
On overlaps $a_i/a_j$ is a regular unit. The integers $-v_P(a_i)$ therefore agree wherever two descriptions apply and define a finite <divisor on an algebraic curve> $D$. Its local description gives $\mathcal L=\mathcal O_X(D)$. A different rational trivialization multiplies all $a_i$ by the same rational function, changing $D$ by a <principal divisor>, so the resulting <divisor class> is intrinsic.
For injectivity, an <isomorphism> $\mathcal O_X(D)\to\mathcal O_X(E)$ becomes multiplication by some $g\in k(X)^*$ on the rational fibre. At each closed point its effect on the local free generators gives
$$
g\,t_P^{-n_P(D)}\mathcal O_{X,P}
=t_P^{-n_P(E)}\mathcal O_{X,P},
\qquad
v_P(g)=n_P(D)-n_P(E).
$$
Hence $D-E=\operatorname{div}(g)$. Conversely this equality gives the required <isomorphism> by multiplication by $g$. We have proved the <group isomorphism>
$$
\boxed{\operatorname{Cl}(X)\xrightarrow{\;\sim\;}
\operatorname{Pic}(X),\qquad[D]\longmapsto[\mathcal O_X(D)].}
$$
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