As I say in my comment above, I don't think you are actually interested in Chern classes of toric varieties. But, if you are:

For a smooth proper toric variety $X$, the $k$-th Chern class of the tangent bundle is represented by the sum of the codimension $k$ toric subvarieties. For example, if $X = \mathbb{P}^2$, then $c_1$ is $3 \cdot [\mathrm{line}]$ and $c_2$ is $3 \cdot [\mathrm{point}]$. I'm having trouble finding you a reference for this, because everyone wants to prove more complicated things! But it isn't hard to prove directly. Let coordinates on the torus be $(t_1, \ldots, t_n)$, for $t_i \in \mathbb{C}^{\ast}$. Let $\theta_i$ be the tangent vector field $t_i \partial/(\partial t_i)$. These extend to sections of $T_{\ast} X$. We'll explicitly take $n-k+1$ sections of $T_{\ast} X$, written as linear combinations of the $\theta_i$ and compute where they become linear dependent.

Let's do $k=n$ first. So we have one section of $T_{\ast} X$, written as $\sum a_i \theta_i$. Let's work in the neighborhood of a fixed point of $X$. Without loss of generality, we may assume that the $t_i$ are local coordinates at that fixed point. So we want to know where $\sum a_i t_i (\partial/\partial t_i)$ vanishes. Well, it's exactly where all the components vanish, so where $a_1 t_1 = a_2 t_2 = \cdots = a_n t_n =0$. Assuming all the $a_i$ are nonzero, this is precisely where $t_1 = \cdots = t_n = 0$. In other words, at the fixed point. The local computation goes the same at every fixed point -- so $c_{n}(T_{\ast} X)$ is represented by the sum of the torus fixed points. Sanity check: The top Chern class of the tangent bundle is the Euler characteristic, and the Euler characteristic of a toric variety is equal to the number of torus fixed points.

Now let's do $k=1$. So we want to take $n$ sections, of the form $\sum_{i=1}^n a_{ij} \theta_i$, and figure out where they are linearly independent. Again, compute in the neighborhood of a torus fixed point. We want $\det (a_{ij} t_i) =0$, or $\det (a_{ij}) t_1 t_2 \cdots t_n$. If we have chosen the constants $a_{ij}$ generically, this is the same as $t_1 t_2 \cdots t_n$, so the union of the coordinate planes through the fixed point. Again, taking the union over all charts, the sections between linearly independent on the union of the toric divisors.

I won't do the general case, but it isn't much harder than these two.