Kleinian circle packings are fractal limit sets that can be visualized as infinite patterns of mutually tangent circles. In this talk, we will discuss a Diophantine approximation problem on such limit sets: how well a point in the limit set can be approximated by tangent points coming from the packing. We will focus on a Good approximation theorem, a Lagrange-type theorem relating approximation quality to geometric lengths, and epsilon-badly approximable points, whose size is measured by Hausdorff dimension. This talk is based on joint work with Seonhee Lim and Yongquan Zhang.