Edit: Now updated to include reference and slightly more general result.
Edit 2: Includes remark about integrability.
Similar to Francesco Polizzi's answer, there is the following Theorem concerning 6-manifolds.
A closed oriented 6-dimensional manifold $X$ without 2-torsion in $H^3(X,\mathbb{Z})$ admits an almost complex structure. There is a 1-1 correspondence between almost complex structures on $X$ and the integral lifts $W \in H^2(X, \mathbb{Z})$ of $w_2(X)$. The Chern classes of the almost complex structure corresponding to $W$ are given by $c_1 = W$ and $c_2 = (W^2 - p_1(X))/2$.
In fact, a necessary and sufficient condition for the existence of an almost complex structure is that $w_2(X)$ maps to zero under the Bockstein map $H^2(X,\mathbb{Z}_2) \to H^3(X,\mathbb{Z})$.
I think the reason for results such as this and the one mentioned by Francesco is the following. To find an almost complex structure amounts to finding a section of a bundle over $X$ with fibre $F_n=SO(2n)/U(n)$. The obstructions to such a section existing lie in the homology groups $H^{k+1}(X, \pi_k(F_n))$. When $n$ is small I would guess we can compute these homotopy groups and so have a good understanding of the obstructions. For example, in the case mentioned above, n=3, $F_n = \mathbb{CP}^3$ and so the only non-trivial homotopy group which concerns us is $\pi_2 \cong \mathbb{Z}$. This is what leads to the above necessary and sufficient condition concerning 2-torsion. On the other hand when $n$ is large I don't know what $F_n$ looks like, let alone its homotopy groups...
For the proof of the above mentioned result see the article "Cubic forms and complex 3-folds" by Okonek and Van de Ven. (I highly recommend this article, it's full of interesting facts about almost complex and complex 3-folds.)
It is worth pointing out that in real dimension 6 or higher there is no known obstruction to the existence of an integrable complex structure. In other words, there is no known example of a manifold of dimension 6 or higher which has an almost complex structure, but not a genuine complex structure. By the classification of compact complex surfaces, those 4-manifolds admitting integrable complex structures are well understood.
This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. You can have a look at
On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958
http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607
but there are many other papers by the same authors.
For a nice and compact exposition, you can look at these class notes of Christian Schnell:
http://www.math.sunysb.edu/~cschnell/pdf/notes/kodaira.pdf
Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Goldman & Millson, Gerstenhaber, Stasheff, Deligne, Quillen, Kontsevich.
Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions,
for $I$ and $I_t$:
$$ T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t $$
and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually
$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.$$
In local coordinates,
$$ \phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k}, $$
and $T^{0,1}_t$ is generated (over the smooth functions) by
$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}. $$
Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$,
$X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures
on $X\times \Delta$, which you can compare.
ADDENDUM
I second YangMills' suggestion to have a look at Chapter 2 of Gross-Huybrechts-Joyce.
You can also try Chapter 1 of K. Fukaya's book "Deformation Theory, Homological algebra, and Mirror Symmetry", as well as the Appendix to Homotopy invariance of the Kuranishi Space by Goldman and Millson (Illinois J. of Math, vol.34, No.2, 1990). In particular, you'll see how one uses formal Kuranishi theory to avoid dealing with the convergence of the power series for $\phi(t)$. For deformations of compact complex manifolds, the convergence was proved by Kodaira-Nirenberg-Spencer. Fukaya says a little bit about the convergence of this series in general, i.e., for other deformation problems.
Best Answer
Turning comments into answer: An example of a closed 6-manifold not admitting an almost complex structure is $S^1 \times (SU(3)/SO(3))$. From the obstruction theory for lifting the map $M \to BSO(6)$ classifying the tangent bundle of an oriented 6-manifold through $BU(3) \to BSO(6)$, one sees that the unique obstruction to the existence of an almost complex structure is the Bockstein of $w_2(M)$, also known as the third integral Stiefel-Whitney class $W_3(M)$. This is generally, in all dimensions, the unique obstruction for an orientable manifold to admit what is called a spin$^c$ structure.
The manifold $SU(3)/SO(3)$ does not admit a spin$^c$ structure (see e.g. Friedrich's "Dirac operators in Riemannian geometry" p.50). Also, a calculation shows that for orientable manifolds $M$ and $N$, the product $M\times N$ is spin$^c$ if and only if each factor is. Hence $S^1 \times (SU(3)/SO(3))$ is not spin$^c$, and thus not almost complex.
To create a simply connected example, one can surger out e.g. any circle of the form $S^1 \times pt$ in $S^1 \times (SU(3)/SO(3))$; note that $SU(3)/SO(3)$ is simply connected as can be seen from the homotopy long exact sequence for $SU(3)/SO(3) \to BSO(3) \to BSU(3)$. The process of taking a manifold, crossing with a circle, and then surgering out such a circle, is sometimes referred to as spinning the original manifold. Spinning a closed orientable manifold produces a spin$^c$ manifold iff the original manifold was spin$^c$, see Proposition 2.4 here https://arxiv.org/abs/1805.04751.
To create more examples for free, you can use the fact that the connected sum $M\# N$ of orientable manifolds is spin$^c$ iff each factor is.