Algebraic de Rham Cohomology of Projective Space over $\mathbb{C}$

In what follows, always assume $X$ to be a Riemann surface - so that one should expect ${\mathbb H}^2 (X,\Omega^\bullet)$ to be Hodge theory's $ H^{(1,1)}$.

Version 1: In the complex topology, $(\Omega^\bullet,\partial)$ is quasi-isomorphic to the de Rahm complex of sheaves $({\cal A}^\bullet, d)$ (with $d = \partial +\overline\partial$): on the one hand, we have the $d$-Poincare lemma; on the other, locally we can integrate any holomorphic differential form to obtain a holomorphic function - which is good enough for Riemann surfaces. [According to Griffith and Harris, p 448, the corresponding statement ($\partial$-Poincare) is true in higher dimensions/degrees.]

Be that as it may, we may therefore calculate the algebraic de Rham using de Rham's de Rham... Write ${\cal A}^k= \sum_ {p+q =k} {\cal A}^{(p,q)}$, where ${\cal A}^{(p,q)} $ is the (soft) sheaf of $(p,q)$ differential forms, and write $d = \partial + \overline\partial$. The spectral sequence (filtration on the $p$) degenerates at the $E_1$ term, with $E_1^{p,q} = H^q_{\overline\partial}(A^{(p,\bullet)})$, where $A^{(p,q)}$ are the global sections of ${\cal A}^{(p,q)}$. Hence, in the case of $X$ a Riemann surface, one has the identification $ {\mathbb H}^2(X, \Omega^{\bullet}_{alg}) = H^1_{\overline\partial}(A^{(1,\bullet)} )= H^1(X, \Omega^1)$. In this analytic/calculus setting, one can choose as generator for $H^{(1,1)}$ the Chern class of the line bundle ${\cal O}(\infty)$, or the Fubini-Study Kaehler form...

Version 2 ( more algebraic, and/but weaker): If $C$ is a complex, write $C[k]$ for the same complex, but shifted so that $C[k]^n = C^{k+n}$. Then, in the case of a curve $X$, on has an exact sequence of (sheaf) complexes $$ 0\to\Omega^1[-1]\to \Omega^{\bullet}_{alg} \to {\cal O} \to 0,$$ with the 'abuse of notation' of identifying a one-term complex with its corresponding term. (Note that this makes sense as a sequence of complexes - it wouldn't make sense to swap the outer [non-zero] terms.)

Taking (hyper-)cohomology gives the long exact sequence $$ \cdots \to H^1(X, {\Omega}^1[1])\to {\mathbb H}^1(X, \Omega^{\bullet}_{alg}) \to H^1(X, {\cal O}) \to \\ H^2(X, \Omega^1[-1]) \to {\mathbb H}^2(X, \Omega^{\bullet}_{alg}) \to H^2(X, {\cal O}) \to \cdots.$$

But, on the one hand, $H^k(X, \Omega^1[-1]) = H^{k-1}(X, \Omega^1)$. On the other, in the case of $X={\mathbb P}^1(\mathbb C)$, $H^1(X, {\cal O})$ and $H^2(X,{\cal O})$ vanish. [Once again, using Hodge theory, we wouldn't need to rely on this]. Therefore one can identify $$ {\mathbb H}^2(X, \Omega^{\bullet}_{alg}) = H^1(X, \Omega^1) .$$ So, with $X = {\mathbb P}^1({\mathbb C})$, in terms of Cech cohomology, $H^1(X, \Omega^1)$ is spanned by (the class of) $ dz/z$ on the open set ${\mathbb P^1}({\mathbb C}) \setminus \{0, \infty \}$.

In the above, I was taking the hypercohomology of a complex of sheaves $\cal C$ to be (up to unique isomorphism) the homology of the complex of modules of global sections $\Gamma(X, {\cal I})$, where ${\cal I}$ is a complex of injectives quasi-isomorphic to ${\cal C}$.

This answer is obviously not very satisfactory. Still, hoping it was of some help... If someone feels like cleaning this up, or improving/rewriting it, rather than writing their own answer, I am happy to make this a community answer.

Comment/Edit: it's worth notating explicitly that G-H's $\partial$-Poincare statement ${\cal H}^k(\Omega^{\bullet}) =0$ for $k>0$ does NOT hold in the Zariski topology. For instance $dz / z $ does not have an algebraic anti-derivative on any open set.

For a sheaf $\mathcal{F}$ on $\mathbb{P}^1$ and an open $U$ in $\mathbb{P}^1$, write $\mathcal{F}_U$ for the sheaf that takes an open $V$ to $\mathcal{F}(U \cap V)$. (This is $\iota_* \mathcal{F}|_U$ in fancy language.)

Let $U = \mathbb{P}^1 \setminus \{\infty\}$ and $V= \mathbb{P}^1 \setminus \{0\}$. Then we get an acyclic resolution of the complex

$0 \to \mathcal{O} \to \ \Omega \to 0$

on $\mathbb{P}^1$ by taking the complex $$ 0 \to \mathcal{O}_U \oplus \mathcal{O}_V \to \mathcal{O}_{U \cap V} \oplus \Omega_U \oplus \Omega_V \to \Omega_{U \cap V} \to 0. $$ All the sheaves involved are acyclic because they're supported on affines. The maps are, respectively: $(f, g) \mapsto (f - g, df, dg)$ and $(\phi, \omega, \eta) \mapsto d\phi + \omega - \eta$.

(Three notes: one, a little abuse of notation in defining these maps, we've suppressed restrictions of a function or form from their domain to a smaller domain from the notation.

Two, every smooth projective curve can be covered by two affines, and the same resolution works for every smooth projective curve.

Three, it is reasonable to ask how you would ever think of this resolution. The point is that we've taken the Cech resolution of $\Omega$ and $\mathcal{O}$ separately, then taken the total complex of the associated double complex. See Kedlaya's article that I left in a comment for more detials, and a proof this always works.)

Let's say we're working in characteristic zero (the definitions make sense in characteristic $p$ but are not typically used since the cohomology theory doesn't satisfy Weil's axioms). Taking global sections and computing homology, we get $$ H^0(\mathbb{P}^1) = \{(f, g) \in \mathcal{O}(U) \oplus \mathcal{O}(V) \ | \ (f - g, df, dg) = (0, 0, 0)\}. $$ Since in characteristic zero $df = 0$ implies $f$ is constant, this shows $H^0$ is one-dimensional generated by $(\lambda, \lambda)$. (Same proof for any smooth projective curve.)

Now the interesting case, $$ H^1(\mathbb{P}^1) = \frac{ \{ (\phi, \omega, \eta) \in \mathcal{O}(U \cap V)\oplus \Omega(U)\oplus \Omega{O}(V)\ | \ d\phi = \omega - \eta\} } { \big\{(\phi, \omega, \eta): \phi \text{ can be written as a difference } f - g \text{ where} f \text{ extends to a function on } U \\ \ \ \ \ \text{ and } g \text{ extends to a function on } V \text{ and } df = \omega \text{ and } dg = \eta \big\}}. $$

This seems verbose but in fact is $0$, which you can see by letting $z$ be a uniformizer at $0$ and writing everything in terms of Laurent polynomials in $z$, then explicitly taking antiderivatives. The key is that you can't get any $dz/z$ terms as these are neither regular at $0$ or $\infty$, but you can antidifferentiate anything else.

Finally, $H^2(\mathbb{P}^1)$ identifies with forms supported on $\Omega_{U \cap V}$ mod exact forms, and mod the image of the restriction maps from $\Omega_U$ or $\Omega_V$. Again using Laurent polynomials, we see this space is generated by the image of $\frac{dz}{z}$.