Otto Forster in his Lectures on Riemann Surfaces calls two Riemann surfaces isomorphic if there exists a biholomorphism between them. Said differently, two Riemann surfaces are (holomorphically) isomorphic if they are isomorphic in the category of Riemann surfaces together with holomorphic maps.
In Forster's terminology, let me show you that any line in $\mathbb{P}^2$ is isomorphic to $\mathbb{P}^1$. With holomorphicity suitably defined, a holomorphic map between Riemann surfaces is continuous. In particular, any line in $\mathbb{P}^2$ is homeomorphic (what you call "topologically isomorphic") to $\mathbb{P}^1$.
Assume without loss of generality that $a\neq 0$. The cases $b\neq 0$ and $c \neq 0$ are treated analogously.
Consider the projection $$\pi\colon X \longrightarrow \mathbb{P}^1 \\ [x:y:z]\longmapsto [y:z].$$ Since $a\neq 0$ holds, we have $[1:0:0]\notin X$. Thus, the mapping $\pi$ is indeed defined on the whole of $X$.
Next consider, as you did, the mapping
$$\iota \colon \mathbb{P}^1 \longrightarrow X \\ [y:z]\longmapsto [-\frac{by+cz}{a}:y:z].$$
Verify that $\iota$ is the two-sided inverse of $\pi$. This makes $\pi$ a bijection.
A bijective holomorphic mapping between Riemann surfaces is biholomorphic by the Local Normal Form of non-constant holomorphic mappings between Riemann surfaces. Thus, it remains to show that $\iota$ (or equivalently $\pi$) is holomorphic.
How to do so depends on your definition of the complex structure on $X$.
If you define the complex structure simply as the one induced from the complex projective line $\mathbb{P}^1$ by the bijection $\pi$, there is of course nothing to show.
Another (a priori different, but ultimately equivalent) definition of a complex structure on $X$, might be phrased along the following lines. Since $a\neq 0$, the homogeneous polynomial $F(x,y,z)=ax+by+cz$ is non-singular. We can thus consider $X$ as a smooth projective plane curve. The complex structure of a general smooth projective plane curve is described in detail on pages 13-16 of Rick Miranda's Algebraic Curves and Riemann Surfaces.
In our case, Miranda's definition amounts to the subsequent complex structure on $X$. Define the two sets $U_1\colon =\{[x:y:z]\in \mathbb{P}^2 \vert \ y\neq 0\}$ and $U_2\colon =\{[x:y:z]\in \mathbb{P}^2 \vert \ z\neq 0\}$. Since $a\neq 0$, we can cover $X$ by the sets $X\cap U_1$ and $X\cap U_2$, which are open in $X$. The charts of $X$ are constructed by employing the Implicit Function Theorem. More explicitly, if $p\in U_1\cap X$, consider the dehomogenization $F(u,1,v)$. Then we have $\frac{\partial F}{\partial u}(u,1,v)=a\neq 0$, and a chart of $X$ near $p$ is given by $\psi_1([x:y:z])=z/y$. Similarly, if $p\in U_2\cap X$, then $\psi_2([x:y:z])=y/z$ defines a chart of $X$ near $p$.
Now, an atlas of $\mathbb{P}^1$ is given by $\phi_1\colon V_1\rightarrow \mathbb{C}; [z:w]\mapsto w/z$ and $\phi_2\colon V_2\rightarrow \mathbb{C}; [z:w]\mapsto z/w$, where $V_1\colon=\{[z:w]\in \mathbb{P}^1\vert z\neq 0\}$ and $V_2\colon=\{[z:w]\in \mathbb{P}^1\vert w\neq 0\}$. The inverses of $\phi_1$ and $\phi_2$ read as $\phi_1^{-1}(w)=[1:w]$ and $\phi_2^{-1}(z)=[z:1]$, respectively. Observe that, wherever it is defined, for $i,j\in \{1,2\}$, the composition $\psi_i\circ \iota \circ \phi^{-1}_j$ is equal either to the holomorphic function $\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}\setminus\{0\};z\mapsto 1/z$ or to the holomorphic function $\mathbb{C}\rightarrow \mathbb{C};z\mapsto z$. Thus, the mapping $\iota$ is holomorphic.
