Let $M$ be a differentiable manifold, $V\in\mathcal{X}(M)$ a vector field on $M$ and $\alpha\in\mathcal{X}^*(M)$ a 1-form. Let $L_{V\alpha}$ be another 1-form defined by: $$L_{V\alpha}(X)=V(\alpha(X))-\alpha([V,X])$$ I need to determine the analytic expression of $L_{V\alpha}$ knowing that, in a given chart $(\mathcal{U},\varphi)$, is, $$V=\sum_iV_i\frac{\partial}{\partial u_i}\text{ and }\alpha=\sum_i \alpha_i du_i$$
I tried to compute $L_{V\alpha}(\frac{\partial}{\partial u_j})$ which, I think, are the "coordinates" of $L_{V\alpha}$ in the base asociated to the chart and I get, $$L_{V\alpha}(\frac{\partial}{\partial u_j})=\sum_i(\frac{\partial \alpha_j}{\partial u_i}V_i+\frac{\partial V_i}{\partial u_j}\alpha_i)$$
I would like to confirm that I got the theory correctly and this is the right method, and also, if it is, I did the computation correctly.
You're almost right, but there's an index error in your last term. It should be $$ \dots =\sum_i\left(\dots +\frac{\partial V_i}{\partial u_j}\alpha_i\right). $$
The $1$-form you're trying to compute is called the Lie derivative of $\alpha$ in the direction $V$. It should be denoted by $L_V(\alpha)$ or $L_V\alpha$: Here, $\alpha$ is meant to be the argument of the Lie derivative operator $L_V$, not a subscript on $V$.