Assuming that $\kappa$ is an **infinite** cardinal, I answer your question with a **characterization.**

For $\mathcal C\subseteq\mathcal P(\kappa),$ the intersection graph $G_\mathcal C=(V,E)$ has the following properties:

(1) $|V|\le2^\kappa;$

(2) there is a collection $\mathcal K$ of cliques, with $|\mathcal K|\le\kappa,$ covering all vertices and edges of the graph; that is, each vertex belongs to an element of $\mathcal K,$ and each pair of adjacent vertices is contained in an element of $\mathcal K.$

**Conversely**, I clain that any graph $G=(V,E)$ satisfying those two conditions is isomorphic to an intersection graph $G_\mathcal C$ with $\mathcal C\subseteq\mathcal P(\kappa).$

Let $G=(V,E)$ be a graph satisfying (1) and (2). We may assume that $V\subseteq\{X:0\in X\subseteq\kappa\}.$ Let $\mathcal K$ be a collection of cliques in $G,$ of cardinality at most $\kappa,$ covering all vertices and edges of $G.$ Since $|\kappa\times\mathcal K|=\kappa,$ it will suffice to show that $G\cong G_\mathcal C$ for some $\mathcal C\subseteq\kappa\times\mathcal K.$

For $v\in V$ define $A_v=\{(\alpha,K)\in\kappa\times\mathcal K:\alpha\in v\in K\}.$

**$v\mapsto A_v\text{ is injective}:$**

Suppose $u,v\in V$ and $\alpha\in u\setminus v.$ Choose $K\in\mathcal K$ with $u\in K;$ then $(\alpha,K)\in A_u\setminus A_v.$

**$\{u,v\}\in E\implies A_u\cap A_v\ne\emptyset$:**

Choose $K\in\mathcal K$ with $u,v\in\mathcal K;$ then $(0,K)\in A_u\cap A_v.$

**For $u\ne v,\ A_u\cap A_v\ne\emptyset\implies\{u,v\}\in E:$**

If $(\alpha,K)\in A_u\cap A_v),$ then $u$ and $v$ are both in the clique $K.$

Therefore $G\cong G_\mathcal C$ where $\mathcal C=\{A_v:v\in V\}.$