Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$?
I know the theoretical approach: for every $t \in [0,1]$ there exist a neighbourhood $U_t$ of $c(t)$ and a local tangent field $C_t \in \mathcal X (U_t)$ that extends $\dot c$, so one computes $[C_t, X]$ and then evaluates it at $c(t)$.
Theory is nice, but putting it into practice does not look obvious; how should I do it? Concretely, the problem is that in local coordinates and choosing $X = \partial _i$ I would obtain $- \sum \limits _j (\partial _i C_t ^j) \partial _j$. How would I evaluate $\partial _i C_t ^j$ in $c(t)$? In other words, how do I compute $\partial _i \dot c ^j$?
The Lie bracket $[\dot c,X]$ is not well defined. A Lie bracket requires two vector fields defined on an open set, so as you noted, to make sense of this bracket you need to extend $\dot c$ to a vector field. But different extensions will give different results.
For example, in $\mathbb R^2$, suppose $c(t) = (t,0)$ and $X(x,y) = \partial/\partial y$. Then $V_1(x,y) = \partial/\partial x$ and $V_2(x,y) = (1+y)\partial/\partial x$ are both extensions of $\dot c$, but $[V_1,X] = 0$ while $[V_2,X] = - \partial/\partial x$.