So for integer spin, the way I understand it mathematically (in a classical limit), is that under a Lorentz transformation (i.e. change of coordinates), spin $n$ particles transform like rank $n$ covariant tensor fields. This makes wonderful sense to me because the induced representation of $SO^+(1,3)$ on $\bigotimes^n \mathbb R^{1,3}$ is given in coordinates by the same rule.
The situation for spin $1/2$ particles is trickier for me to see. I know that generally spinors are elements in a complex vector space $\Delta$, which admits a faithful representation of of the Clifford algebra $CL(t,s)$, called the Dirac representtion. In case of $t=1,s=3$, we have that $\Delta=\mathbb{C}^4$, and the representation is given by the isomorphism $CL(1,3)\otimes \mathbb{C}\rightarrow \text{End}(\mathbb{C}^4)$. We define the double cover $\text{Spin}^+(1,3)\rightarrow SO^+(1,3)$ as subset of $CL(1,3)$, then the spinor representation is the faithful given by the restriction of this isomorphism to $\text{Spin}^+(1,3)$, which has image in $GL(\mathbb{C}^4)$.
What I don't understand is how to characterize how spin $1/2$ particles transform under a Lorentz transformation. In particular, fermions are spinor fields, i.e. maps $\Psi:\mathbb{R}^{t,s}\rightarrow \mathbb{R}^{t,s}\times \Delta$ which satisfy: $$\pi_{\mathbb{R}^{t,s}}\circ \Psi=\text{Id}_{t,s}$$ However, how do we prescribe a representation of $SO^+(1,3)$ on these spinor fields? I know that spinor fields have a natural transformation property related to the spinor representation of the double cover $\text{Spin}^+(1,3)$ on $\mathbb{C}^4$, but how do we translate this into a representation of $SO^+(1,3)$ on $\mathbb{C}^4$ ?
Edit:
Ok I think I get it now, each element in $\text{Spin}^+(t,s)$ maps to an element in $SO^+(1,3)$, so under $\text{Spin}^+(t,s)$, we have that spinor fields then transform under $\text{Spin}^+(t,s)$ by:
$$g\cdot \Psi=g\cdot (x,\psi(x))=(\lambda(g)\cdot x,\kappa(g)\cdot \psi(x))$$
where $\lambda$ is the covering map, and $\kappa$ is the spinor representation. Does that make sense?
