$V=\left\{a\partial_x+bx+cI:a,b,c\in\mathbb{C}\right\}$ with $[X,Y]:=XY-YX$
Question 1. $V$ with $[X,Y]$ is a Lie álgebra with dimension 3, right?
By Ado's theorem, the Lie Álgebra $V$ is isomorphic to a Lie subalgebra of the square matrix space $M(n,\mathbb{R})$ some $n$ (this is what I have been able to understand)
Question 2. What would be the matrix Lie subalgebra that is isomorphic to $V$?
We have answered the question together in the comments. You don't need Ado's theorem.
Let $(e_1,e_2,e_3)=(\partial_x,x,I)$ be a basis of $L$. Then $[e_1,e_2]=e_3$ and $[e_1,e_3]=[e_2,e_3]=0$. Hence $L$ is the Heisenberg Lie algebra, with underlying vector space $V$.
It has a faithful matrix representation $\rho\colon L\hookrightarrow \mathfrak{gl}_3(\Bbb C)$, given by $$ \rho(e_1)={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}},\quad \rho(e_2)={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\\\end{pmatrix}},\quad \rho(e_3)={\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\\\end{pmatrix}}. $$ The image is isomorphic to the Lie subalgebra of strictly upper-triangular matrices.