The scheme $Spec \mathbb{C}[x,y]^{\mathbb{C}^{\times}}=Spec \mathbb{C}$ is a point, so this approach to giving $X/\mathbb{C}^{\times}$ the structure of a variety does not work. The quotient is $\mathbb{P}^1$ in particular it is not an affine variety. In order to realize this, we need to remove the fixed point $(0,0)$ under the $\mathbb{C}^{\times}$ action and this of $X/(0,0)$ as a union $U_1\cup U_2$ where $U_1=Spec \mathbb{C}[x,y,y^{-1}]$ (i.e where $y\neq 0$) and $U_2=Spec \mathbb{C}[x,y,x^{-1}]$ (where $x\neq 0$). The quotients $U_1/\mathbb{C}^{\times}=Spec \mathbb{C}[x,y,y^{-1}]^{\mathbb{C}^{\times}}=Spec \mathbb{C}[x/y]$ and $U_2/\mathbb{C}^{\times}=Spec \mathbb{C}[x,y,x^{-1}]^{\mathbb{C}^{\times}}=Spec \mathbb{C}[y/x]$ glue together to $\mathbb{P}^1$.

If you want to understand the general theory, I'd recommend reading about the GIT construction of quotients. This involves choosing a very ample line bundle $L$ on a variety $X$ such that the group action of $G$ on $X$ lifts to $L$. After removing a certain "bad" subset from $X$ called the locus of non-semistable points $X^{ns}$, the GIT quotient is simply $Proj \left(\bigoplus_{m\geq 0} H^0(X/X^{ns},L^{\otimes m})^{G}\right)$. The group $G$ needs to be reductive for this to work.

In the example you quoted the non-semistable locus is the origin (for the most natural example of $L$ with $\mathbb{C}^{\times}$ action). In general it is a closed subvariety.