 Solution

Choose a nonconstant rational function $t$ on $C$. Functions in $L(D)$ are integral over a finite-dimensional bounded-pole space over $k(t)$; equivalently, evaluation of sufficiently many principal parts embeds $L(D)$ into a finite-dimensional vector space. Hence $L(D)$ is finite dimensional.

If $E=D+\mathrm{div}(h)$, multiplication by $h$ gives the isomorphism $L(E)\to L(D)$. A canonical divisor is the divisor of a nonzero rational differential. Riemann-Roch theorem says $\ell (D)-\ell (K-D)=\mathrm{deg}D+1-g$. Taking $D=0$ gives $1-g=1+g-\mathrm{deg}K$, hence $\mathrm{deg}K=2g-2$.

For a smooth plane curve of degree $d$, the adjunction formula gives $K_C\sim (d-3)H{|}_C$. A cubic has $d=3$, so $K_C\sim 0$.