Clustering is one of the most fundamental problems in unsupervised learning with a large number of applications. However, classical clustering algorithms assume that the data is static, thus failing to capture many real-world applications where data is constantly changing and evolving. Driven by this, we study the metric $k$-center clustering problem in the fully dynamic setting, where the goal is to efficiently maintain a clustering while supporting an intermixed sequence of insertions and deletions of points. This model also supports queries of the form (1) report whether a given point is a center or (2) determine the cluster a point is assigned to. We present a deterministic dynamic algorithm for the $k$-center clustering problem that provably achieves a $(2+ε)$-approximation in poly-logarithmic update and query time, if the underlying metric has bounded doubling dimension, its aspect ratio is bounded by a polynomial and $ε$ is a constant. An important feature of our algorithm is that the update and query times are independent of $k$. We confirm the practical relevance of this feature via an extensive experimental study which shows that for values of $k$ and $ε$ suggested by theory, our algorithmic construction outperforms the state-of-the-art algorithm in terms of solution quality and running time.