Consider $$ \dot{x}=f(x,\alpha), x\in\mathbb{R},\alpha\in\mathbb{R} $$ with smooth $f$ and equilibrium $x=0$ at $\alpha=0$, $\lambda=f_{x}(0,0)=0$ and, moreover, $f_{xx}(0,0)\neq 0, f_{\alpha}(0,0)\neq 0$.
I know that the normal form is given by $$ \dot{x}=\alpha+x^2+O(x^3).~~(*) $$
I would like to understand the derivation of this normal form by starting with the system $\dot{x}=f(x,\alpha), \dot{\alpha}=0$ and using some normal form theory.
I think the idea is to write $$ \dot{x}=f_{\alpha}(0,0)\alpha + g(x,\alpha),~~\dot{\alpha}=0, $$ where $g(x,\alpha)$ is the non-linear part and can be written as $g(x,\alpha)=ax^2+b\alpha x+c\alpha^2+ O(\lvert x\rvert^3+\lvert\alpha\rvert^3)$ where probably some terms are non-resonant. However, since $\alpha$ is supposed to be a (nonzero) constant, I think we can rewrite this as $$ \dot{x}=f_{\alpha}(0,0)\alpha+a_1x+a_2x^2+a_3+O(\lvert x\rvert^3), $$ where $a_1, a_2, a_3$ are coefficients. I think this already is the normal form. However, we can improve/modify it to get $(*)$ :
The first step probably would be to get rid of the linear term $a_1x+a_3$ by coordinate transformation $X=x+\delta$.
Then, we would have $\dot{x}=f_{\alpha}(0,0)\alpha+a_1x^2+O(\lvert x\rvert^3)$. I guess the next step would consist of two rescalings in order to replace $f_{\alpha}(0,0)$ by $1$ and $a_1$ by $1$. Probably for this, we need the conditions that $f_{\alpha}(0,0)\neq 0$ and $f_{xx}(0,0)\neq 0$.
Is this the correct way of thinking?
This is generally the correct way of thinking. However, you can avoid a lot of steps by directly using the coordinate transformation you're planning to use, $X = x + \delta$. Moreover, since you're primarily interested in removing the $\alpha x$ term, you already know that the variable transformation has to be of the form \begin{equation} x = \xi + \delta \alpha. \end{equation} Then, you can expand $f(\xi+\delta \alpha,\alpha)$ around $\xi = 0$, $\alpha = 0$ and choose $\delta$ such that the $\alpha \xi$ term vanishes. In the process, you'll find where the conditions $f_\alpha(0,0) \neq 0$ and $f_{xx}(0,0) \neq 0$ come into play.