Consider a dataset where data is collected on multiple features of multiple individuals over multiple times. This type of data can be represented as a three dimensional individual/feature/time tensor and has become increasingly prominent in various areas of science. The tensor biclustering problem computes a subset of individuals and a subset of features whose signal trajectories over time lie in a low-dimensional subspace, modeling similarity among the signal trajectories while allowing different scalings across different individuals or different features. We study the information-theoretic limit of this problem under a generative model. Moreover, we propose an efficient spectral algorithm to solve the tensor biclustering problem and analyze its achievability bound in an asymptotic regime. Finally, we show the efficiency of our proposed method in several synthetic and real datasets.