Current approaches to evaluating large language models (LLMs) with pre-existing Knowledge Graphs (KG) mostly ignore the structure of the KG and make arbitrary choices of which part of the graph to evaluate. In this paper, we introduce KGLens, a method to evaluate LLMs by generating natural language questions from a KG in a structure aware manner so that we can characterize its performance on a more aggregated level. KGLens uses a parameterized KG, where each edge is augmented with a beta distribution that guides how to sample edges from the KG for QA testing. As the evaluation proceeds, different edges of the parameterized KG are sampled and assessed appropriately, converging to a more global picture of the performance of the LLMs on the KG as a whole. In our experiments, we construct three domain-specific KGs for knowledge assessment, comprising over 19,000 edges, 700 relations, and 21,000 entities. The results demonstrate that KGLens can not only assess overall performance but also provide topic, temporal, and relation analyses of LLMs. This showcases the adaptability and customizability of KGLens, emphasizing its ability to focus the evaluation based on specific criteria.