Continuous-time Markov chains on non-negative integers can be used for modeling biological systems, population dynamics, and queueing models. Qualitative behaviors of birth-and-death models, typical examples of such one-dimensional continuous-time Markov chains, have been substantially studied. For one-dimensional Markov chains with polynomial transition rates, recent studies provided criteria for their long-term behavior. In this paper, we provide a classification of Markov chains on non-negative integers when the transition rates are arbitrary functions. The criteria are written with asymptotics of the transition rates. This classification implies their dynamical properties, including explosivity, recurrence, positive recurrence, and exponential ergodicity. As an application, we derive a complete classification (if and only if conditions) for those dynamical features when the transition rates have certain expansion forms, which include all rational functions, so that our classifications cover mass-action kinetics, Michaelis-Menten kinetics, and Haldane equations. Our classification solely relies on easily computable quantities: the maximal degree of the expansion of the transition rates, the mean and the variance of the transition rates. We demonstrate the utility of this classification framework using the approximation of high-dimensional mass-action systems by a one-dimensional reaction system with general rational kinetics.