A possible method for getting explicit representatives involves Griffiths's Residues. More in general, let $X \subset \mathbb{P}^{n+1}$ a smooth hypersurface of degree $d$.
Denote by $H^n_{\textrm{prim}}(X)$ the primitive subspace of the n-th cohomology of $X$, that is $$ H^n_{\textrm{prim}}(X)= \textrm{Ker} ( \lambda: H^n(X) \to H^{n+2}(X) ),$$ $$\lambda(\alpha)= \alpha \cup c_1 (\mathcal{O}_X(1)).$$
The (restriction on) the Hodge filtration $F^p H^n_{\textrm{prim}}(X)$ can be identified with $$H^0(\Omega^{n+1}_{\mathbb{P}}(n-p+1)(X)) / d H^0((\Omega^{n}_{\mathbb{P}}(n-p)(X)).$$

Now, if $$\omega= \sum_i (-1)^i x_i dx_0 \wedge \ldots d\hat{x_i} \wedge \ldots dx_{n+1}$$ write $$H^0(\Omega^{n+1}_{\mathbb{P}}(k))=\big\{ \frac{g \omega}{f^k}\ | \ g \in \mathbb{C}[x_0, \ldots, x_{n+1}]_{kd-n-2} \big \} $$
and use this identification in the quotient above.

One has that if $R_f:= \mathbb{C}[x_0, \ldots, x_{n+1}]/J_f$ with $J_f$ being the ideal generated by the partial derivatives of $f$ then $$H^{n-p,p}_{\textrm{prim}}(X):= H^n_{\textrm{prim}}(X) \cap H^{n-p,p}(X) \cong (R_f)_{(p+1)d-n-2},$$
where the maps sends the class of $g\omega/f^k$ to the class of $g$.

Now this is all a very nice bit of theory, but in this case the situation is very simple. For a curve we have $H^1(C, \mathbb{C}) \cong H^{1,0}(C) \oplus H^{0,1}(C)$ and moreover $H^1_{\textrm{prim}}(C) \cong H^1(C, \mathbb{C})$ (and the same for the subspaces of course).
The degree of the equation is three, therefore one has $H^0(K_C) \cong (R_f)_0$ and $H^1(\mathcal{O}_C) \cong (R_f)_3$. The former is generated by 1 (it is canonically identified with $\mathbb{C}$). For the latter, your equation for $C$ implies that you can pick as generator for $(R_f)_3 \cong \mathbb{C}\cong \langle z^3 \rangle$. Note that $R_3$ is the socle of the Jacobian ring, that is $(R_f)_t=0$ for $t>3$.
Moreover (in the non weighted case) if $\rho$ is the degree corresponding to the socle of this (artinian, Gorenstein) ring one has $R_a \cong (R_{\rho-a})^*$, and this is in turn just another expression of Serre Duality.

Plugging these polynomials in the quotient above, one gets an explicit expression as desired.