A good concise reference is Deligne's article on the Weil conjectures for K3 surfaces.  See "La conjecture de Weil pour les surfaces K3" in Inventiones 15 (1971/72): 206-226.  An English version is easy to Google too.  In Deligne, and in other sources, the group GSpin is called CSpin.  Here are the pertinent details.

Let $(V,q)$ be a finite-dimensional vector space with nondegenerate quadratic form, over any field (any base scheme is fine, adapting definitions appropriately... see the comment by nfdc23 below).  Let $C(V,q)$ be the Clifford algebra, and $C^+$ the even part of the Clifford algebra.  Embed $V$ in $C(V,q)$ as the degree 1 part; thus for $v \in V$, we view $v \in C(V,q)$ and $v \cdot v = q(v)$.  

Then the group $CSpin$ is defined: 
$$CSpin(V,q) = \{ g \in C^+ : g V g^{-1} = V \}.$$ 
From above, we have a natural homomorphism $CSpin(V,q) \rightarrow SO(V,q)$ sending $g \in CSpin(V,q)$ to the map $(v \mapsto g v g^{-1})$.  The kernel consists of scalars, giving the short exact sequence
$$1 \rightarrow G_m \rightarrow CSpin(V,q) \rightarrow SO(V,q) \rightarrow 1.$$

The group $Spin(V,q)$ is the kernel of the spinor norm $CSpin(V,q) \rightarrow G_m$.  Altogether, this gives the diagram used by Deligne:
$$
\begin{array}{ccccc}
& & Spin(V,q)  & & \\
& & \downarrow & \searrow & \\
G_m & \rightarrow & CSpin(V,q) & \rightarrow & SO(V,q) \\
& \searrow & \downarrow & & \\
& & G_m & & \\
\end{array}
$$
(Excuse the pooly formatted diagram!)  The bottom-left diagonal arrow is the map $x \mapsto x^{-2}$.

This pretty quickly gives an identification 
$$CSpin(V,q) \cong \frac{Spin(V,q) \times G_m}{\mu_2}.$$
Here $\mu_2$ is identified with the kernel of $Spin(V,q) \rightarrow SO(V,q)$.

If you want to describe this $\mu_2$ using coroots, the only subtlety is in type $D_n$ with $n$ even, i.e., in $Spin_{4n}$.  (Thanks to Mikhail Borovoi for the correction and nfdc23 for the suggestion.)  Outside of $n=4$, the central $\mu_2$ in $Spin(V,q)$ should be the only one fixed by the Dynkin diagram automorphism.  In type $D_4$, i.e., for $Spin_8$, the representation $Spin(V,q) \rightarrow SO(V,q)$ determines a node on the Dynkin diagram.  The central $\mu_2$ in $Spin(V,q)$ is the only one fixed by the Dynkin diagram automorphism fixing that node.

This makes it pretty easy to relate the root datum of a (split) $CSpin$ group to that of the simply-connected $Spin$ group (which is its derived subgroup). 

For simply-connected groups, one can look up the full root datum in any good reference (e.g., Bourbaki).