To keep things simple, let us assume we work over a perfect field.
The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group $\mathbb G_m=\mathbb A^1-\{0\}$ has the $\mathbb A^1$-homotopy type of the infinite dimensional projective space. Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as $[X,B\mathbb G_m]=\text{Pic}(X)=CH^1(X)$, where $[?,?]$ stands for the $\text{Hom}$ in the $\mathbb A^1$-homotopy category of $k$-schemes, denoted by $H(k)$.

In general, we denote by $K(\mathbb Z(n),2n)$ the $n$-th motivic Eilenberg-MacLane space, i.e. the object of $H(k)$ which represents the $n$-th Chow group in $H(k)$: for any smooth $k$-scheme $X$, one has

$$[X,\Omega^i K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n))$$

(where $\Omega^i$ stands for the $i$-th loop space functor).

Then, there are several models for $K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that $K(\mathbb Z(1),2)$ is the infinite projective space.
$K(\mathbb Z(0),0)$ is simply the constant sheaf $\mathbb Z$. For higher $n$, here is a construction (this is Voevodsky's).

Given a $k$-scheme $X$, denote by $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite correspondences from $V$ to $X$ (i.e. the finite linear combinations of cycles
$\sum n_iZ_i$ in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in $X$ (I will need this functoriality only for closed immersions).

Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line.
The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of $K(\mathbb Z(n),2n)$ in $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i):L(Y)\to L(X)$.

If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of $\mathbb P^1$-spectra, but maybe it is enough at first to think of $\mathbb P^1$-spectra simply as the cohomology theories
allowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum $MU$ which represents complex cobordism in algebraic topology). The idea is that $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of $\mathbb P^1$ with coefficients in $E(1)$, is the same as a map of ring spectra $MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories ($MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is $KGL$, the $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if $k$ is of characteristic zero, but is not published yet, and it is known for any field $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).