According to Milne's Algebraic Groups, it is possible to define an algebraic $K$-group as a functor from a small category of finitely generated $K$-algebras to the category of groups. I am interested in defining an algebraic group which would play the role of $\mathbb Z/2\mathbb Z$ (although I am not sure what that sentence means).
Using this functorial approach, I would set
$$R\mapsto R^+/2R^+,$$
where $R^+$ denotes the underlying abelian group structure of $R$ and $2R^+$ is the image of the morphism $r\mapsto 2r$.
The first question is: Is this definition correct?
If so, then: What is the schematic description of this algebraic group?
First of all, you forget one important issue in the definition: What you define called a group functor we call it a $K$- group if it is representable, i. e., if there is a scheme $Y$ over $K$ such that your functor send $R$ to $Hom(Spec\, R,Y)$. Not every functor is representable. For example, it is easy to see that the group functor $R\to \mathbb{Z}/\mathbb{2Z}$ is not representable: If it were representable by $Y$, we should have $$\mathbb{Z}/\mathbb{2Z}=Hom(Spec\,k\times k,Y)=Hom(Spec\, k,Y)\times Hom(Spec k,Y)=\mathbb{Z}/\mathbb{2Z}\times\mathbb{Z}/\mathbb{2Z},$$ which is a contradiction.
Second, you want an analog of $\mathbb{Z}/\mathbb{2Z}$, but what is $\mathbb{Z}/\mathbb{2Z}$? It is the unique group such that for every other group $G$, $Hom(\mathbb{Z}/\mathbb{2Z},G)$ is the $2$-torsion of $G$. so you have to find a group with a similar property. The group functor we defined in the last paragraph was a good candidate, but it is not a $K$-group scheme. But if you look at the problem, you can see that the problem is that the schemes can be disconnected. The same problem happens in the definition of the constant sheaf. The solution is simple: Let's denote by $n_R$ the number of connected components of $Spec\,R$ and define a group functor $R\to (\mathbb{Z}/\mathbb{2Z})^{n_R}$.
It is easy to see that this functor is representable by $Y=Spec (K\times K)$. Gor the group structure, keep in mind that $Y$ has two points and they should be associated with the two elements of $\mathbb{Z}/\mathbb{2Z}$, so define the coproduct $$K\times K\to (K\times K)\otimes (K\times K)$$ to be the map $$(1,0)\to (1,0)\otimes (1,0)+(0,1)\otimes(0,1),(0,1)\to (1,0)\otimes (0,1)+(0,1)\otimes (1,0)$$
This group is called the constant group scheme $\mathbb{Z}/\mathbb{2Z}$ and has the universal property we want: It can detect the $2$-torsion of group schemes.