It isn't too bad to describe the irreducible representations of $SO(4)$. We can realize $SO(4)$ as $SU(2) \times SU(2) / \langle (- \text{Id}, - \text{Id})\rangle$. To see this, identify $\mathbb{R}^4$ with the quaternions, and identify $SU(2)$ with the norm $1$ quaternions. For $(u,v) \in SU(2) \times SU(2)$, the map $q \mapsto uqv^{-1}$ from the quaternions to themselves preserves the norm, so we get a map $SU(2) \times SU(2) \to SO(4)$; one can check that this map is surjective and has kernel the two element group generated by $(- \text{Id}, - \text{Id})$.

The irreducible representations of $SU(2)$ are well known to be $\text{Sym}^m(\mathbb{C}^2)$ for $m \geq 0$. This has dimension $m+1$, and $- \mathrm{Id}$ acts by $(-1)^m$. So the irreps of $SU(2) \times SU(2)$ are $\text{Sym}^m(\mathbb{C}^2) \boxtimes \text{Sym}^n(\mathbb{C}^2)$, with dimension $mn$, and with $(- \text{Id}, - \text{Id})$ acting by $(-1)^{m+n}$. This representation factors through $SO(4)$ if and only if $m+n$ is even, so the irreps of $SO(4)$ are $\text{Sym}^m(\mathbb{C}^2) \boxtimes \text{Sym}^n(\mathbb{C}^2)$ for $m+n \equiv 0 \bmod 2$. Whenever $m \equiv n \equiv 0 \bmod 2$, then the kernel contains the $4$-dimensional subgroup $(\pm\text{Id}, \pm\text{Id})$, so the smallest faithful representation is $\mathbb{C}^2 \boxtimes \mathbb{C}^2$, with dimension $4$.