I think that your intuition is correct, but the question is a bit subtle. FIrst of all, there are topological obstructions for the group of biholomorphisms to be a complex Lie group, as pointed out in the comment by Andrew Hwang. One simple case in which such obstructions vanish (and to be honest, the only one I ever studied in any detail) is when the manifold is compact.
So, assume for the moment we have a compact manifold $X$, together with a complex structure $J$. I will denote by $\operatorname{Aut}(X,J)$ the group of biholomorphisms. A $1$-parameter subgroup, say $\{f_t\mid t\in\mathbb{R}\}\subset\operatorname{Aut}(X,J)$ gives you a vector field $v$ on $X$ by setting $$v_p:=\partial_{t=0}f_t(p).$$ Notice that this is just a real vector field, i.e. a section of $TX\to X$. As $f_t^*J=J$, we see however that $v$ satisfies \begin{equation}\label{realhol}
\tag{$\star$}
\mathcal{L}_vJ=0
\end{equation}
and for any $v$ that satisfies \eqref{realhol}, its flow $f_t$ will be a subgroup of $\operatorname{Aut}(X,J)$ - the flow will be complete, as $X$ is compact. I guess we can already see that this might be a problem for noncompact manifolds.
Sections of $TX$ that satisfy \eqref{realhol} are usually called real-holomorphic vector fields. From this point of view then, the lie algebra of $\operatorname{Aut}(X,J)$ is the algebra of real-holomorphic vector fields. Notice however that:
if $v$ is real-holomorphic, also $Jv$ is;
$v$ is real-holomorphic if and only if $v^{1,0}$ is holomorphic, meaning that it is a holomorphic section of the holomorphic vector bundle $T^{1,0}X$.
So we can identify the Lie algebra of $\operatorname{Aut}(X,J)$ either with holomorphic or real-holomorphic vector fields, and it has a complex structure coming either from remark $1$ above, if you like to use real-holomorphic vector fields, or just from multiplication by $i$, if you prefer holomorphic vector fields. However, we are still missing an useful tool, as one might wonder what the complex analogue of the flow of a real-holomorphic vector field is.
So, let $v^{1,0}$ be a holomorphic vector field, and consider the real-holomorphic vector fields $v$ and $Jv$. notice that
\begin{equation}
v=2\,\Re(v^{1,0})$\mbox{ and }Jv=-2\,\Im(v^{1,0})
\end{equation}
and notice also that by \eqref{realhol}, $[v,Jv]=0$. So the flows of $v$ and $Jv$ commute; call them $f^v_t$ and $f^{Jv}_s$, and consider the composition
\begin{equation}
\Phi_{z}:=f^v_t\circ f^{Jv}_s,\ \mbox{ for }z=t+\operatorname{i}s
\end{equation}
Then $\Phi_z$ is a subgroup of $\operatorname{Aut}(X,J)$, and it can be checked that for any $p\in M$, $$v^{1,0}_p=\partial_{z=0}\Phi_z(p).$$
In other words, $\Phi_z$ is the "complex flow" of the holomorphic vector field $v^{1,0}$.
Best Answer
We can define holomorphic sections of any holomorphic vector bundle in the same way as we define holomorphic functions. Let $X$ be a complex manifold and let $E \to X$ be a holomorphic vector bundle over $X$. We can extend the $\overline\partial$ to act on sections of $E$: Let $E_U \to U \times \mathbb C^r$ be a local trivialization and $(e_1, \dots, e_r)$ be a local holomorphic frame of $E$. If $\sigma = \sum_j s_j e_j$ is a section of $E$ over $U$, then we set $$ \overline\partial \sigma := \sum_j \overline \partial s_j \otimes e_j. $$ If $E_V \to V \times \mathbb C^r$ is another trivialization, then we write $g(z,\lambda) = (z, g(z) \lambda)$ for the induced transition function. These are holomorphic, so $g(z)$ is a $r \times r$ matrix of holomorphic functions. If we write $\sigma_U$ and $\sigma_V$ for the representations of the section $\sigma$ in the frames over $U$ and $V$, then $\sigma_U = g \sigma_V$. It follows that $$ \overline \partial \sigma_U = g \overline \partial \sigma_V $$ because $g$ is holomorphic, so the $\overline \partial$ operator glues to define an operator on the space of sections of $E$.
We now define holomorphic sections of $E$ to be smooth sections $\sigma$ such that $\overline \partial \sigma = 0$. If we pick a local holomorphic frame $(e_1, \dots, e_r)$ and write $\sigma = \sum_j s_j e_j$ as before, then this entails that $\sigma$ is holomorphic if and only if all the functions $s_j$ are holomorphic.
We could of course have defined holomorphic sections as being those sections that satisfy that the "coordinate functions" $s_j$ are holomorphic in any local holomorphic frame. Since the transition functions of $E$ are holomorphic, this is well defined. This is basically the same as what we did here.
Since you ask for additional resources for dealing with holomorphic tangent fields specifially, I encourage you to have a look at the Bochner--Weitzenböck formulas you asked about on MO the other day. These are often used to show that there are no non-zero holomorphic vector fields on a manifold (a fun exercise is to prove this by using the Kähler--Einstein metric on a projective manifold with ample canonical bundle -- try Ballmann or Zheng's books if you need help on this).