Graphene-based heterostructures are a promising material system for designing the topologically nontrivial Chern insulating devices. Recently, a two-dimensional (2D) monolayer ferromagnetic insulator CrI$_{3}$ was successfully synthesized in experiments [Huang et al., Nature 546, 270 (2017)]. Here, these two interesting materials are proposed to build a heterostructure (Gr/CrI$_{3}$). Our first-principles calculations show that the system forms a van der Waals (vdW) heterostructure, relatively facilely fabricated in experiments. A Chern insulating state is acquired in the Gr/CrI$_{3}$ heterostructure if the vdW gap is compressed to certain extents by applying an external pressure. Amazingly, very strong magnetization (about 150 meV) is found in graphene, induced by the substrate CrI$_{3}$, despite the vdW interactions between them. A low-energy effective model is employed to understand the mechanism. The work functions, contact types, and band alignments of the Gr/CrI$_{3}$ heterostructure system are also studied. Our work demonstrates that the Gr/CrI$_{3}$ heterostructure is a promising system to observe the quantum anomalous Hall effect at high temperatures (up to 45 K) in experiments.
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